## Key Concepts

- Solving equations with variable on both sides
- Solving equation with infinitely many or no solutions
- Use equations to solve problems

## Solving algebraic equations

If an equation has pronumerals on both sides, collect the like terms to one side by adding or subtracting terms.

**Example:**

2x+1 = 5x−4

1 = 3x−4 [Subtracting 2x from both sides]

3x = 5 [Adding 4 to both sides]

x = 5/3 [Dividing by 3]

### Solve equations using algebra tiles

**Example: **Solve for 4x + 7 = 2(2x + 1) + 5

**Solution:** Let us represent the equation using algebra tiles

Since 7=7 is a true statement, the equation is true for all values of x.

So, x can have **infinitely many solutions**.

### Identity

An equation that is true for all values of the variable is an **identity**.

**Q: **Solve 4x – 3 = 2(2x – 3) + 3

**Sol: **4x – 3 = 4x – 6 + 3

4x – 3 = 4x – 3

– 3 = – 3

### Equation that has no solution

**Q:** Find the value of x if 3(x + 2) – 7 + 2x = 5x + 4.

**Sol:** 3(x + 2) – 7 + 2x = 5x + 4

3x + 6 – 7 + 2x = 5x + 4 [Distributive property]

(3x + 2x) + (6 – 7) = 5x + 4 [Add like terms]

5x – 1 = 5x + 4 [Maintain the equality by subtracting 5x from both sides]

– 1 = 4

There is no value of x that makes the equation true.

Therefore, the equation has **no solution**.

## Exercise

- Solve 5x-4 = 4x
- The triangle shown is isosceles. Find the length of the third side of the triangle.

- Solve the equation – 3(8+3h) = 5h+4
- Find the missing value in – 2(2x- ?) + 1 = 17-4x
- Is the equation – 4(3-2x) = -12-8x an identity?

### Concept Map

### What have we learned

- If an equation has pronumerals on both sides, collect the like terms to one side by adding or subtracting terms.

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