** Key Concepts**

- Relate Solutions of Linear Systems
- Estimate Solutions of systems by inspection
- Estimate more solutions of systems of inspection

**Estimate Solutions By Inspection**

- The value of
*m*in the equation y= mx + b represents the ……………

- When lines are the same distance apart over their entire lengths, they are …………

- The ……………. is the value
*b*in the equation y=mx+ b.

- A……………… is a relationship between two variables that gives a straight line when graphed.

- Identify the slope and y-intercept of the equation y=2x-3. Slope………..and y-intercept…….

- How many solutions does this system have? y = x-3 and 4x 10y = 6

**Answers**

- Slope

- Parallel

- Y-intercept

- Linear equation

- 2 and -3

- I don’t know. Ok let’s find out.

Solve & Discuss It!

Draw three pairs of lines, each showing a different way that two lines can intersect or not intersect. How are these pairs of lines related?

#### By these graphs, we can find the solution of the equation.

How are slopes and y-intercepts related to the number of solutions of a system of linear equations?

Ok, let us know ……….

**Relate Solutions of Linear Systems**

**Essential Question **

How are slopes and y-intercepts related to the number of solutions of a system of linear equations?

### Example 1:

Deanna drew the pairs of lines below. Each pair of lines represents a system of linear equations.

How can you use the graphs to determine the number of solutions of each system?

- A system of
is formed by two or more linear equations that use the same variables.*linear equations*

We can solve a system of linear equations

- By the graph, the points of intersection of the graphed lines relate to the solutions of the systems of linear equations
- By comparing with slope and y-intercept form y= mx+c.

**Case1: **

The equations of the linear system y= x+4 and y= – x + 6.

y= x + 4

y= – x + 6

Look for relationships

How do the points of intersection of the graphed lines relate to the solutions of the systems of linear equations?

Compare with slope-intercept form y = m x + b

The equations of the linear system

y = 1.x + 4 ………….. y = m_{1}x+ b_{1}

y = – 1.x + 6 …………….. y = m_{2}x + b_{2}

First line slope is m_{1} = 1

Second line slope is m_{2} = -1

m_{1 }≠ m_{2}

They have different slopes.

The system has 1 solution (1, 5).

**Case 2:**

The equations of the linear system y= x + 3 and y = x + 1

y =x+3

y =x+1

Compare with slope-intercept form y = m x + b

The equations of the linear system

y =1. x + 3 ………… y = m_{1}x +b_{1}

y =1. x + 1 ………… y =m_{2}x +b_{2}

First line slope is m_{1} =1 y-intercept b_{1}=3

Second line slope is m_{2 }=1 y-intercept b_{2}=1

m_{1 }= m_{2}=1 b_{1}≠ b_{2}

The equations of the linear system have the same slopes and different y-intercepts.

The system has no solution.

**Case 3:**

The equations of the linear system x + y= -2 and 3x + 3y =- 6

x+y=-2 or y =-x-2

3x+3y= -6 or y =-x-2

The lines intersect at every point; they are the same line.

This system has infinitely many solutions.

Compare with slope-intercept form y = m . x + b

The equations of the linear system x + y= -2 and 3x + 3y =- 6

The equations of the linear system

y = – 1.x – 2 ………… y =m_{1}x +b_{1}

y =−3x−63 −3x−63 or y = -1x – 2 ………… y =m_{2}x +b_{2}

First line slope is

m_{1}=-1 y-intercept b_{1}=-2

Second line slope is

m_{2}=-1 y-intercept b_{2}=-2

m_{1} = m_{2} = -1 b1 = b2= −2

The equations of the linear system have the same slopes and same y-intercepts.

They represent the same line. The system has infinitely many solutions.

You can inspect the slopes and y-intercepts of the equations in a system of linear equations in order to determine the number of solutions of the system.

**Estimate Solutions by Inspection**

### Example 2:

Harrison and Pia each buy *x *comic books. Harrison also buys an action figure for $15, while Pia buys a different action figure for $12. Could they each spend the same amount, *y*, and buy the same number of comic books? Explain.

Solution:

The system of equations represents the situation.

y = 5 x + 15

y = 5 x + 12

The slopes are the same. The y-intercepts are different.

The system has no solution.

Harrison and Pia could not spend the same amount of money and buy the same number of comic books.

**Estimate More Solutions of Systems by Inspection**

### Example 3:

Corey and Winnie each bought x pounds of cheddar cheese and y pounds of tomatoes. Corey spent $12 at the supermarket. Winnie spent $24 at the farmer’s market. Could they have bought the same amount by weight of cheddar cheese and tomatoes? Explain.

**Solution:**

The system of equations 6x + 2y = 12

12x + 4y = 24 represents the situation.

Write each equation in slope-intercept form.

First equation

б x + 2y = 12

2у = -б х + 12

у = – 3.x +6

Second equation

12х + 4y = 24

4y =- 12х + 24

у= -3. х +6

The equations represent the same line. Every (x, y) pair on the line is a solution.

Corey and Winnie bought the same amount of cheese and tomatoes.

Let us multiply the first equation 6x + 2y = 12 by 2

2(6x + 2y = 12)

12x + 4y = 24

If one linear equation is a multiple of another, the equations represent the same line, and the system of equations has infinitely many.

### Practice & Problem Solving

- The equations represent the heights
*y*, of the flowers, in inches, after*x*days.

What does the y-intercept of each equation represent? Will the flowers ever be the same height? Explain.

**Solution:**

The system of linear equations y= 0.7x + 2 and

y= 0.4x + 2.

Compare with slope and y-intercept form

The slopes are not equal, and the y-intercepts are equal.

The lines intersect at one point. They have one solution.

Y-intercept of each equation represents the initial heights of the flowers.

The flowers will be the same height only at the start**.**

2. Maia says that the two lines in this system of linear equations are parallel. Is she correct? Explain. 2x+y=14 2y +4x=14

**Solution: **

Given linear equations are 2x+y=14 and 2y+4x=14

Convert the equations into slope-intercept form.

2x+y=14 2y+4x=14

y=14-2x 2y = 14 – 4x

y = -2x + 14 2y= – 4 x+14

y=-2x+7

Slope-intercept form of given equations

y = -2 x + 14

y =-2 x + 7

The slopes are the same. The y-intercepts are different.

### Let’s check your knowledge

- How are slopes and y-intercepts related to the number of solutions of a system of linear equations?

- How many solutions does this system have?

y = x-3 4x – 10y =6

- How many solutions does this system have?

x + 3y = 0 12y =- 4x

- Determine whether this system of equations has one solution, no solution, or infinitely many solutions.

y = 8x + 2 y = -8x + 2

- Does this system have one solution, no solution, or an infinite number of solutions?

4x + 3y= 8 8x + 6y = 2

#### Answers:

- Systems of equations can be solved by looking at their graphs.

- A system with one solution has one point of intersection.
- A system with no solution has no points of intersection.
- A system with infinitely many solutions has infinite points of intersection.

or

- The slopes are different. The lines intersect at 1 point.
- The slopes are the same, and the y intercepts are different. The lines are parallel.
- The slopes are the same, and the y intercepts are the same. The lines are the same.

- Given equations

y = x – 3 and 4x – 10y=6

**Step 1:**

Write slope-intercept form of second equation.

**Step 2:**

Compare both equations with slope-intercept form

**Given equations**

The slopes are different. The lines intersect at 1 point. The system has one solution.

**Given equations**

y = 8x + 2 y = -8x + 2

Compare with slope-intercept form y=mx + b.

Slopes are 8 and -8

The slopes are different. The lines intersect at 1 point.

So, this system of equations has one solution.

**Given equations**

4x + 3y= 8 8x + 6y = 2

Write slope–intercept form of both equations.

The slopes are the same, and the y- intercepts are different. The lines are parallel.

This system has one solution, no solution.

### Key Concept

You can inspect the slopes and y-intercepts of the equations in a system of linear equations in order to determine the number of solutions of the system.

**Concept Map **

**What Have We Learned**

- Understand how to find the number of solutions of a system of equations by inspecting

the equations. - Understand how to solve systems of two linear equations… by inspection.
- Understand how to relate solutions of Linear Systems.
- Understand how to Estimate Solutions of Systems by Inspection
- Understand how to Estimate More Solutions of Systems by Inspection

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