Question

# What expression represents the total area of the four white triangles?

Hint:

### The methods used to find the product of binomials are called special products.

Multiplying a number by itself is often called squaring.

For example (*x* + 3)(*x* + 3) = (*x* + 3)2

Area of a square = (side)2

## The correct answer is: 12x + 36

### The area of the outer square of side x+6 cm = (x+6)^{2}

(x+6)^{2} = (x+6)(x+6) = x(x+6) +6(x+6)

= x(x) + x(6) +6(x) +6(6)

= x^{2} + 6x + 6x + 36

= x^{2} + 12x + 36

The area of the inner square of side x cm = x^{2}

Now, Total area of four white triangles = Area of the outer square - area of the inner square

= x^{2} + 12x + 36 - x^{2}

= 12x + 36

Final Answer:

Hence, the expression for the total area of the four white triangles is 12x + 36.

^{2}= (x+6)(x+6) = x(x+6) +6(x+6)

^{2}+ 6x + 6x + 36

^{2}+ 12x + 36

The area of the inner square of side x cm = x

^{2}

Now, Total area of four white triangles = Area of the outer square - area of the inner square

^{2}+ 12x + 36 - x

^{2}

Final Answer:

Hence, the expression for the total area of the four white triangles is 12x + 36.

### Related Questions to study

### Write the product in the standard form. (𝑥^{2} − 2𝑦)(𝑥^{2} + 2𝑦)

^{2}− 2y)(x

^{2}+ 2y) = x

^{2}(x

^{2}+ 2y) - 2y(x

^{2}+ 2y)

= x^{2}(x^{2}) + x^{2}(2y) - 2y(x^{2}) - 2y(2y)

= x^{4} + 2x^{2}y - 2x^{2}y - 4y^{2}

= x^{4} - 4y^{2}

Final Answer:

Hence, the simplified form of (𝑥^{2} − 2𝑦)(𝑥^{2} + 2𝑦) is x^{4} - 4y^{2}.

### Write the product in the standard form. (𝑥^{2} − 2𝑦)(𝑥^{2} + 2𝑦)

^{2}− 2y)(x

^{2}+ 2y) = x

^{2}(x

^{2}+ 2y) - 2y(x

^{2}+ 2y)

= x^{2}(x^{2}) + x^{2}(2y) - 2y(x^{2}) - 2y(2y)

= x^{4} + 2x^{2}y - 2x^{2}y - 4y^{2}

= x^{4} - 4y^{2}

Final Answer:

Hence, the simplified form of (𝑥^{2} − 2𝑦)(𝑥^{2} + 2𝑦) is x^{4} - 4y^{2}.

### Write the product in the standard form. (𝑥 − 2.5)(𝑥 + 2.5)

= x(x + ) - (x + )

= x(x) + x() - (x) - ()

= x^{2} + x - x -

= x^{2} -

= x^{2} - 6.25

Final Answer:

Hence, the simplified form of (𝑥 − 2.5)(𝑥 + 2.5) is x^{2} - 6.25.

### Write the product in the standard form. (𝑥 − 2.5)(𝑥 + 2.5)

= x(x + ) - (x + )

= x(x) + x() - (x) - ()

= x^{2} + x - x -

= x^{2} -

= x^{2} - 6.25

Final Answer:

Hence, the simplified form of (𝑥 − 2.5)(𝑥 + 2.5) is x^{2} - 6.25.

### Describe the possible lengths of the third side of the triangle given the lengths of the other two sides.

5 inches, 12 inches

- Hints:

- Triangle inequality theorem
- According to this theorem, in any triangle, sum of two sides is greater than third side,
- a < b + c

c < a + b

- while finding possible lengths of third side use below formula

- Step-by-step explanation:

- Given:

a = 5 inches, b = 12 inches.

- Step-by-step explanation:

- Given:

a = 5 inches, b = 12 inches.

- Step 1:
- Find length of third side.

c < a + b

∴ c < 5 + 12

c < 17

- Step 1:
- Find length of third side.

c < a + b

∴ c < 5 + 12

c < 17

- Step 2:

- Step 2:

12 – 5 < c < 5 + 12

7 < c < 17

Hence, all numbers between 7 and 17 will be the length of third side.

- Final Answer:

### Describe the possible lengths of the third side of the triangle given the lengths of the other two sides.

5 inches, 12 inches

- Hints:

- Triangle inequality theorem
- According to this theorem, in any triangle, sum of two sides is greater than third side,
- a < b + c

c < a + b

- while finding possible lengths of third side use below formula

- Step-by-step explanation:

- Given:

a = 5 inches, b = 12 inches.

- Step-by-step explanation:

- Given:

a = 5 inches, b = 12 inches.

- Step 1:
- Find length of third side.

c < a + b

∴ c < 5 + 12

c < 17

- Step 1:
- Find length of third side.

c < a + b

∴ c < 5 + 12

c < 17

- Step 2:

- Step 2:

12 – 5 < c < 5 + 12

7 < c < 17

Hence, all numbers between 7 and 17 will be the length of third side.

- Final Answer:

### Write the product in the standard form. (3𝑎 − 4𝑏)(3𝑎 + 4𝑏)

= 3a(3a) + 3a(4b) - 4b(3a) - 4b(4b)

= 9a

^{2}+ 12ab - 12ab - 16b

^{2}

= 9a

^{2}- 16b

^{2}

Final Answer:

Hence, the simplified form of (3𝑎 − 4𝑏)(3𝑎 + 4𝑏) is 9a

^{2}- 16b

^{2}.

### Write the product in the standard form. (3𝑎 − 4𝑏)(3𝑎 + 4𝑏)

= 3a(3a) + 3a(4b) - 4b(3a) - 4b(4b)

= 9a

^{2}+ 12ab - 12ab - 16b

^{2}

= 9a

^{2}- 16b

^{2}

Final Answer:

Hence, the simplified form of (3𝑎 − 4𝑏)(3𝑎 + 4𝑏) is 9a

^{2}- 16b

^{2}.

### Write the product in the standard form.

Final Answer:

Hence, the simplified form of .

**Write the product in the standard form. **

**Maths-General**

Final Answer:

Hence, the simplified form of .

**Maths-Explanation:**

it is possible that the sum of two quadratic trinomials is a linear binomial

If the other terms get cancel

Example:

,

On addition we will get , which is a linear binomial.Maths-GeneralExplanation:

it is possible that the sum of two quadratic trinomials is a linear binomial

If the other terms get cancel

Example:

,

On addition we will get , which is a linear binomial.Maths-### If (3x-4) (5x+7) = 15x

Answer:

○ Two terms.

(3x-4) (5x+7) = 15x

○ Step 1:

○ Simplify the right side:

○ (3x-4) (5x+7)

3x(5x+7) - 4(5x+7)

15x

15x

○ Step 1:

○ compare both side:

15x

By comparing we get

a = -1

### If (3x-4) (5x+7) = 15x

Maths-GeneralAnswer:

○ Two terms.

(3x-4) (5x+7) = 15x

○ Step 1:

○ Simplify the right side:

○ (3x-4) (5x+7)

3x(5x+7) - 4(5x+7)

15x

15x

○ Step 1:

○ compare both side:

15x

By comparing we get

a = -1

Maths-### The difference of x

Answer:

○ Always take like terms together while performing subtraction.

○ In addition to polynomials only terms with the same coefficient are subtracted.

One polynomial: x

Difference: x

○ Step 1:

○ Let another polynomial be A.

So,

(x

A = (x

A = x

A = x

A = x

### The difference of x

Maths-GeneralAnswer:

○ Always take like terms together while performing subtraction.

○ In addition to polynomials only terms with the same coefficient are subtracted.

One polynomial: x

Difference: x

○ Step 1:

○ Let another polynomial be A.

So,

(x

A = (x

A = x

A = x

A = x

Maths-83 can be written as (90 - 7) and 97 can be written as (90 + 7)

So, 83 × 97 can be written (90 - 7) × (90 + 7)

(90 - 7) × (90 + 7) = 90(90 + 7) - 7(90 + 7)

Maths-General83 can be written as (90 - 7) and 97 can be written as (90 + 7)

So, 83 × 97 can be written (90 - 7) × (90 + 7)

(90 - 7) × (90 + 7) = 90(90 + 7) - 7(90 + 7)

Maths-Hint:

Gradient is also called the slope of the line. The slope intercept form of the equation of the line is y = mx + c, where m is the slope of the line and c is the y-intercept. First we convert the given equation in this form. Further, we compare the equation with the standard form to get the slope and the y-intercept.

Step by step solution:

The given equation of the line is

4x + y = -10

We need to convert this equation in the slope-intercept form of the line, which is

y = mx + c, where m is the slope of the line and c is the y – intercept.

Rewriting the given equation, that is, keeping only the term containing y in the left hand side, we get

y = -4x - 10

Comparing the above equation with y = mx + c, we get

m = -4 ;c = -10

Thus, we get

Gradient = -4

y-intercept = -10

Note:

We can find the slope and y-intercept directly from the general form of the equation too; slope = and y-intercept = , where the general form of equation of a line is ax + by + c=0. Using this method, be careful to check that the equation is in general form before applying the formula.Maths-GeneralHint:

Gradient is also called the slope of the line. The slope intercept form of the equation of the line is y = mx + c, where m is the slope of the line and c is the y-intercept. First we convert the given equation in this form. Further, we compare the equation with the standard form to get the slope and the y-intercept.

Step by step solution:

The given equation of the line is

4x + y = -10

We need to convert this equation in the slope-intercept form of the line, which is

y = mx + c, where m is the slope of the line and c is the y – intercept.

Rewriting the given equation, that is, keeping only the term containing y in the left hand side, we get

y = -4x - 10

Comparing the above equation with y = mx + c, we get

m = -4 ;c = -10

Thus, we get

Gradient = -4

y-intercept = -10

Note:

We can find the slope and y-intercept directly from the general form of the equation too; slope = and y-intercept = , where the general form of equation of a line is ax + by + c=0. Using this method, be careful to check that the equation is in general form before applying the formula.Maths-32 can be written as (30 + 2) and 28 can be written as (30 - 2)

So, 32 × 28 can be written (30 + 2) × (30 - 2)

(30 + 2) × (30 - 2) = 30(30 - 2) + 2(30 - 2)

Maths-General32 can be written as (30 + 2) and 28 can be written as (30 - 2)

So, 32 × 28 can be written (30 + 2) × (30 - 2)

(30 + 2) × (30 - 2) = 30(30 - 2) + 2(30 - 2)

Maths-### The sum of two expressions is x

Answer:

○ Always take like terms together while performing addition.

○ In subtraction of polynomials only coefficients are subtracted.

Sum: x

Term: x

○ Step 1:

○ Let the other term be A.

As given sum is x

A + x

A = x

A = x

A = x

A = x

### The sum of two expressions is x

Maths-GeneralAnswer:

○ Always take like terms together while performing addition.

○ In subtraction of polynomials only coefficients are subtracted.

Sum: x

Term: x

○ Step 1:

○ Let the other term be A.

As given sum is x

A + x

A = x

A = x

A = x

A = x

Maths-### Use the square of a binomial to find the value. 72

72

(70 + 2)(70 + 2) = 70(70 + 2) + 2(70 + 2)

### Use the square of a binomial to find the value. 72

Maths-General72

(70 + 2)(70 + 2) = 70(70 + 2) + 2(70 + 2)

Maths-Hint:

The slope/ gradient of a line is the measure of steepness of a line. It is understood that the slope of all parallel lines in the xy plane are equal. So first, we find the slope from the given equation of a line by using the slope intercept form of a line which is y = mx + c, , where m is slope and c is the y intercept. This gradient will be equal to the gradient of any line parallel to it.

Step by step solution:

The given equation of the line is

-2x + y = -7

We convert this equation in the slope intercept form, which is

y = mx + c

Where m is the slope of the line and c is the y-intercept.

We rewrite the equation -2x + y - 7, as below

y = 2x - 7

Comparing with y = mx + c, we get that m = 2

Thus, the gradient of line -2x + y = 7 is m = 2.

We know that the gradient of any two parallel lines in the xy plane is always equal.

Hence, the gradient of a line parallel to the line whose equation -2x + y = -7 is m = 2.

Note:

We can find the slope and y-intercept directly from the general form of the equation too; slope = and y-intercept = , where the general form of equation of a line is ax + by + c = 0. Using this method, be careful to check that the equation is in general form before applying the formula. Here, we have, a = -2, b = 1, so we getMaths-GeneralHint:

The slope/ gradient of a line is the measure of steepness of a line. It is understood that the slope of all parallel lines in the xy plane are equal. So first, we find the slope from the given equation of a line by using the slope intercept form of a line which is y = mx + c, , where m is slope and c is the y intercept. This gradient will be equal to the gradient of any line parallel to it.

Step by step solution:

The given equation of the line is

-2x + y = -7

We convert this equation in the slope intercept form, which is

y = mx + c

Where m is the slope of the line and c is the y-intercept.

We rewrite the equation -2x + y - 7, as below

y = 2x - 7

Comparing with y = mx + c, we get that m = 2

Thus, the gradient of line -2x + y = 7 is m = 2.

We know that the gradient of any two parallel lines in the xy plane is always equal.

Hence, the gradient of a line parallel to the line whose equation -2x + y = -7 is m = 2.

Note:

We can find the slope and y-intercept directly from the general form of the equation too; slope = and y-intercept = , where the general form of equation of a line is ax + by + c = 0. Using this method, be careful to check that the equation is in general form before applying the formula. Here, we have, a = -2, b = 1, so we getMaths-Answer:

The area of the rectangle is the product of sides.

Two sides of a rectangle

(3p+5q) units and ( 5p-7q ) units.

○ Step 1:

We know, the area of rectangle is product of its sides

i.e. area = side × side

So,

Area = (3p+5q) × (5p-7q)

= 3p (5p -7q) + 5q(5p-7q)

= 15p

= 15p

Maths-GeneralAnswer:

The area of the rectangle is the product of sides.

Two sides of a rectangle

(3p+5q) units and ( 5p-7q ) units.

○ Step 1:

We know, the area of rectangle is product of its sides

i.e. area = side × side

So,

Area = (3p+5q) × (5p-7q)

= 3p (5p -7q) + 5q(5p-7q)

= 15p

= 15p

### How is it possible that the sum of two quadratic trinomials is a linear binomial?

- We have to find out how is it possible that the sum of two quadratic trinomials is a linear binomial.

it is possible that the sum of two quadratic trinomials is a linear binomial

If the other terms get cancel

Example:

,

On addition we will get , which is a linear binomial.

### How is it possible that the sum of two quadratic trinomials is a linear binomial?

- We have to find out how is it possible that the sum of two quadratic trinomials is a linear binomial.

it is possible that the sum of two quadratic trinomials is a linear binomial

If the other terms get cancel

Example:

,

On addition we will get , which is a linear binomial.

### If (3x-4) (5x+7) = 15x^{2}-ax-28, so find the value of a?

- Hint:

- Step by step explanation:

○ Two terms.

(3x-4) (5x+7) = 15x

^{2}-ax-28

○ Step 1:

○ Simplify the right side:

○ (3x-4) (5x+7)

3x(5x+7) - 4(5x+7)

15x

^{2 }+ 21x - 20x - 28

15x

^{2 }+ x - 28

○ Step 1:

○ compare both side:

15x

^{2 }+ x - 28 =15x

^{2}- ax - 28

By comparing we get

a = -1

- Final Answer:

### If (3x-4) (5x+7) = 15x^{2}-ax-28, so find the value of a?

- Hint:

- Step by step explanation:

○ Two terms.

(3x-4) (5x+7) = 15x

^{2}-ax-28

○ Step 1:

○ Simplify the right side:

○ (3x-4) (5x+7)

3x(5x+7) - 4(5x+7)

15x

^{2 }+ 21x - 20x - 28

15x

^{2 }+ x - 28

○ Step 1:

○ compare both side:

15x

^{2 }+ x - 28 =15x

^{2}- ax - 28

By comparing we get

a = -1

- Final Answer:

### The difference of x^{4}+2x^{2}-3x+7 and another polynomial is x^{3}+x^{2}+x-1. What is the

another polynomial?

- Hint:

○ Always take like terms together while performing subtraction.

○ In addition to polynomials only terms with the same coefficient are subtracted.

- Step by step explanation:

One polynomial: x

^{4}+2x

^{2}-3x+7

Difference: x

^{3}+x

^{2}+x-1.

○ Step 1:

○ Let another polynomial be A.

So,

(x

^{4}+2x

^{2}-3x+7) – A = (x

^{3}+x

^{2}+x-1)

A = (x

^{4}+2x

^{2}-3x+7) - (x

^{3}+x

^{2}+x-1)

A = x

^{4 }+ 2x

^{2 }- 3x + 7 - x

^{3 }- x

^{2 }- x + 1

A = x

^{4}- x

^{3 }+ 2x

^{2}- x

^{2 }- 3x - x + 7 + 1

A = x

^{4 }- x

^{3 }+ x

^{2 }- 4x + 8

- Final Answer:

^{4 }- x

^{3 }+ x

^{2 }- 4x + 8.

### The difference of x^{4}+2x^{2}-3x+7 and another polynomial is x^{3}+x^{2}+x-1. What is the

another polynomial?

- Hint:

○ Always take like terms together while performing subtraction.

○ In addition to polynomials only terms with the same coefficient are subtracted.

- Step by step explanation:

One polynomial: x

^{4}+2x

^{2}-3x+7

Difference: x

^{3}+x

^{2}+x-1.

○ Step 1:

○ Let another polynomial be A.

So,

(x

^{4}+2x

^{2}-3x+7) – A = (x

^{3}+x

^{2}+x-1)

A = (x

^{4}+2x

^{2}-3x+7) - (x

^{3}+x

^{2}+x-1)

A = x

^{4 }+ 2x

^{2 }- 3x + 7 - x

^{3 }- x

^{2 }- x + 1

A = x

^{4}- x

^{3 }+ 2x

^{2}- x

^{2 }- 3x - x + 7 + 1

A = x

^{4 }- x

^{3 }+ x

^{2 }- 4x + 8

- Final Answer:

^{4 }- x

^{3 }+ x

^{2 }- 4x + 8.

### Use the product of sum and difference to find 83 × 97.

So, 83 × 97 can be written (90 - 7) × (90 + 7)

(90 - 7) × (90 + 7) = 90(90 + 7) - 7(90 + 7)

= 90(90) + 90(7) - 7(90) - 7(7)

= 8100 + 630 - 630 - 49

= 8100 - 49

= 8051

Final Answer:

Hence, the simplified form of 83 × 97 is 8051.

### Use the product of sum and difference to find 83 × 97.

So, 83 × 97 can be written (90 - 7) × (90 + 7)

(90 - 7) × (90 + 7) = 90(90 + 7) - 7(90 + 7)

= 90(90) + 90(7) - 7(90) - 7(7)

= 8100 + 630 - 630 - 49

= 8100 - 49

= 8051

Final Answer:

Hence, the simplified form of 83 × 97 is 8051.

### Determine the gradient and y-intercept from the following equation: 4x + y = -10

Gradient is also called the slope of the line. The slope intercept form of the equation of the line is y = mx + c, where m is the slope of the line and c is the y-intercept. First we convert the given equation in this form. Further, we compare the equation with the standard form to get the slope and the y-intercept.

Step by step solution:

The given equation of the line is

4x + y = -10

We need to convert this equation in the slope-intercept form of the line, which is

y = mx + c, where m is the slope of the line and c is the y – intercept.

Rewriting the given equation, that is, keeping only the term containing y in the left hand side, we get

y = -4x - 10

Comparing the above equation with y = mx + c, we get

m = -4 ;c = -10

Thus, we get

Gradient = -4

y-intercept = -10

Note:

We can find the slope and y-intercept directly from the general form of the equation too; slope = and y-intercept = , where the general form of equation of a line is ax + by + c=0. Using this method, be careful to check that the equation is in general form before applying the formula.

### Determine the gradient and y-intercept from the following equation: 4x + y = -10

Gradient is also called the slope of the line. The slope intercept form of the equation of the line is y = mx + c, where m is the slope of the line and c is the y-intercept. First we convert the given equation in this form. Further, we compare the equation with the standard form to get the slope and the y-intercept.

Step by step solution:

The given equation of the line is

4x + y = -10

We need to convert this equation in the slope-intercept form of the line, which is

y = mx + c, where m is the slope of the line and c is the y – intercept.

Rewriting the given equation, that is, keeping only the term containing y in the left hand side, we get

y = -4x - 10

Comparing the above equation with y = mx + c, we get

m = -4 ;c = -10

Thus, we get

Gradient = -4

y-intercept = -10

Note:

We can find the slope and y-intercept directly from the general form of the equation too; slope = and y-intercept = , where the general form of equation of a line is ax + by + c=0. Using this method, be careful to check that the equation is in general form before applying the formula.

### Use the product of sum and difference to find 32 × 28.

So, 32 × 28 can be written (30 + 2) × (30 - 2)

(30 + 2) × (30 - 2) = 30(30 - 2) + 2(30 - 2)

= 30(30) + 30(-2) + 2(30) + 2(-2)

= 900 - 60 + 60 - 4

= 900 - 4

= 896

Final Answer:

Hence, the simplified form of 32 × 28 is 896.

### Use the product of sum and difference to find 32 × 28.

So, 32 × 28 can be written (30 + 2) × (30 - 2)

(30 + 2) × (30 - 2) = 30(30 - 2) + 2(30 - 2)

= 30(30) + 30(-2) + 2(30) + 2(-2)

= 900 - 60 + 60 - 4

= 900 - 4

= 896

Final Answer:

Hence, the simplified form of 32 × 28 is 896.

### The sum of two expressions is x^{3}-x^{2}+3x-2. If one of them is x^{2 }+ 5x - 6, what is the

other?

- Hint:

○ Always take like terms together while performing addition.

○ In subtraction of polynomials only coefficients are subtracted.

- Step by step explanation:

Sum: x

^{3}-x

^{2}+ 3x- 2

Term: x

^{2}+ 5x- 6

○ Step 1:

○ Let the other term be A.

As given sum is x

^{3}-x

^{2}+ 3x- 2

A + x

^{2}+ 5x- 6 = x

^{3}-x

^{2}+ 3x- 2

A = x

^{3}-x

^{2}+ 3x- 2 ) - ( x

^{2}+ 5x- 6 )

A = x

^{3}-x

^{2}+ 3x - 2 - x

^{2}- 5x + 6

A = x

^{3}-x

^{2 }- x

^{2}+ 3x - 5x - 2 + 6

A = x

^{3}-2x

^{2}- 2x + 4

- Final Answer:

^{3}-2x

^{2}- 2x + 4.

### The sum of two expressions is x^{3}-x^{2}+3x-2. If one of them is x^{2 }+ 5x - 6, what is the

other?

- Hint:

○ Always take like terms together while performing addition.

○ In subtraction of polynomials only coefficients are subtracted.

- Step by step explanation:

Sum: x

^{3}-x

^{2}+ 3x- 2

Term: x

^{2}+ 5x- 6

○ Step 1:

○ Let the other term be A.

As given sum is x

^{3}-x

^{2}+ 3x- 2

A + x

^{2}+ 5x- 6 = x

^{3}-x

^{2}+ 3x- 2

A = x

^{3}-x

^{2}+ 3x- 2 ) - ( x

^{2}+ 5x- 6 )

A = x

^{3}-x

^{2}+ 3x - 2 - x

^{2}- 5x + 6

A = x

^{3}-x

^{2 }- x

^{2}+ 3x - 5x - 2 + 6

A = x

^{3}-2x

^{2}- 2x + 4

- Final Answer:

^{3}-2x

^{2}- 2x + 4.

### Use the square of a binomial to find the value. 72^{2}

^{2}can be written as (70 + 2)

^{2}which can be further written as (70 + 2)(70 + 2)

(70 + 2)(70 + 2) = 70(70 + 2) + 2(70 + 2)

= 70(70) + 70(2) + 2(70) + 2(2)

= 4900 + 140 + 140 + 4

= 4900 + 280 + 4

= 5184

Final Answer:

Hence, the value of 72^{2} is 5184.

### Use the square of a binomial to find the value. 72^{2}

^{2}can be written as (70 + 2)

^{2}which can be further written as (70 + 2)(70 + 2)

(70 + 2)(70 + 2) = 70(70 + 2) + 2(70 + 2)

= 70(70) + 70(2) + 2(70) + 2(2)

= 4900 + 140 + 140 + 4

= 4900 + 280 + 4

= 5184

Final Answer:

Hence, the value of 72^{2} is 5184.

### What is the gradient of a line parallel to the line whose equation -2x + y = -7 is:

The slope/ gradient of a line is the measure of steepness of a line. It is understood that the slope of all parallel lines in the xy plane are equal. So first, we find the slope from the given equation of a line by using the slope intercept form of a line which is y = mx + c, , where m is slope and c is the y intercept. This gradient will be equal to the gradient of any line parallel to it.

Step by step solution:

The given equation of the line is

-2x + y = -7

We convert this equation in the slope intercept form, which is

y = mx + c

Where m is the slope of the line and c is the y-intercept.

We rewrite the equation -2x + y - 7, as below

y = 2x - 7

Comparing with y = mx + c, we get that m = 2

Thus, the gradient of line -2x + y = 7 is m = 2.

We know that the gradient of any two parallel lines in the xy plane is always equal.

Hence, the gradient of a line parallel to the line whose equation -2x + y = -7 is m = 2.

Note:

We can find the slope and y-intercept directly from the general form of the equation too; slope = and y-intercept = , where the general form of equation of a line is ax + by + c = 0. Using this method, be careful to check that the equation is in general form before applying the formula. Here, we have, a = -2, b = 1, so we get

### What is the gradient of a line parallel to the line whose equation -2x + y = -7 is:

The slope/ gradient of a line is the measure of steepness of a line. It is understood that the slope of all parallel lines in the xy plane are equal. So first, we find the slope from the given equation of a line by using the slope intercept form of a line which is y = mx + c, , where m is slope and c is the y intercept. This gradient will be equal to the gradient of any line parallel to it.

Step by step solution:

The given equation of the line is

-2x + y = -7

We convert this equation in the slope intercept form, which is

y = mx + c

Where m is the slope of the line and c is the y-intercept.

We rewrite the equation -2x + y - 7, as below

y = 2x - 7

Comparing with y = mx + c, we get that m = 2

Thus, the gradient of line -2x + y = 7 is m = 2.

We know that the gradient of any two parallel lines in the xy plane is always equal.

Hence, the gradient of a line parallel to the line whose equation -2x + y = -7 is m = 2.

Note:

We can find the slope and y-intercept directly from the general form of the equation too; slope = and y-intercept = , where the general form of equation of a line is ax + by + c = 0. Using this method, be careful to check that the equation is in general form before applying the formula. Here, we have, a = -2, b = 1, so we get

### Two sides of a rectangle are (3p+5q) units and ( 5p-7q ) units. What is its area?

- Hint:

The area of the rectangle is the product of sides.

- Step by step explanation:

Two sides of a rectangle

(3p+5q) units and ( 5p-7q ) units.

○ Step 1:

We know, the area of rectangle is product of its sides

i.e. area = side × side

So,

Area = (3p+5q) × (5p-7q)

= 3p (5p -7q) + 5q(5p-7q)

= 15p

^{2}- 21pq + 25pq - 35q

^{2}

= 15p

^{2}+ 4pq - 35q

^{2}sq. units

- Final Answer:

^{2}+ 4pq - 35q

^{2}sq. units.

### Two sides of a rectangle are (3p+5q) units and ( 5p-7q ) units. What is its area?

- Hint:

The area of the rectangle is the product of sides.

- Step by step explanation:

Two sides of a rectangle

(3p+5q) units and ( 5p-7q ) units.

○ Step 1:

We know, the area of rectangle is product of its sides

i.e. area = side × side

So,

Area = (3p+5q) × (5p-7q)

= 3p (5p -7q) + 5q(5p-7q)

= 15p

^{2}- 21pq + 25pq - 35q

^{2}

= 15p

^{2}+ 4pq - 35q

^{2}sq. units

- Final Answer:

^{2}+ 4pq - 35q

^{2}sq. units.

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