Determining A Solution By Inspection

 Key Concepts

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

Estimate Solutions By Inspection

1. The value of m in the equation y= mx + b represents the …………… 

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

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

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

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

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

Answers 

1. Slope 

2. Parallel 

3. Y-intercept 

4. Linear equation 

5. 2 and -3 

6. 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 linear equations is formed by two or more linear equations that use the same variables. 

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 = m₁x+ b₁

y = – 1.x + 6 ……………..   y = m₂x + b₂

First line slope is m₁ = 1 

Second line slope is m₂ = -1 

m₁ ≠ m₂

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 = m₁x +b₁

 y =1. x + 1     ………… y =m₂x +b₂

First line slope is m₁ =1                             y-intercept b₁=3   

Second line slope is m₂ =1                        y-intercept b₂=1 

m₁ = m₂=1                                          b₁≠ b₂

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 =m₁x +b₁

 y =−3x−63 −3x−63  or y = -1x – 2        ………… y =m₂x +b₂

First line slope is

m₁=-1                      y-intercept b₁=-2   

Second line slope is

m₂=-1                 y-intercept b₂=-2 

m₁ = m₂ = -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 

1. 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 

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

2. How many solutions does this system have?  

y = x-3     4x – 10y =6  

3. How many solutions does this system have?  

x + 3y = 0             12y =- 4x  

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

y = 8x + 2     y = -8x + 2 

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

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

Answers: 

1. 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. 

2. 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  

3. Given equations 

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

4. 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. 

5. 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