1. 5.1 Solving Systems of Linear Equations by Graphing

2. System of Linear Equations
A set of two or more linear equations in the same variables
Example: x + y = 7 Equation 1
2x – 3y = Equation 2
Solution of a system of Linear Equations
In two variables, is an ordered pair that is the solution to each equation in the system

3. Example 1: Checking Solutions
Tell whether the ordered pair is a solution of the system of linear equations.
(2,5); x + y = Equation 1
2x – 3y = Equation 2
Substitute 2 in for x and 5 in for y in each equation.
Equation 1
x + y = 7
2 + 5 = 7
7 = 7
Equation 2
2x – 3y = -11
2(2) – 3(5) = -11
4 – 15 = -11
-11 = -11
Since the ordered pair (2,5) is a solution of each of the equations, it is a solution to the linear system.

4. Example 1: Checking Solutions
Tell whether the ordered pair is a solution of the system of linear equations. b) (-2,0); y = -2x – 4 Equation 1 y = x + 4 Equation 2 Substitute -2 in for x and 0 in for y in each equation.
Equation 1
y = -2x - 4
0 = -2(-2) - 4
0 = 4 - 4
0 = 0
Equation 2
y = x + 4
0 =
0 = 2
Since the ordered pair (-2,0) is not a solution of each of the equations, it is NOT a solution to the linear system.

5. You try!
Tell whether the ordered pair is a solution to the system of linear equations.
(1, -2); 2x + y = ) (1,4); y = 3x + 1
-x + 2y = y = -x + 5
Equation 1
2(1) + (-2) = 0
2 + (-2) = 0
0 = 0
Equation 1
4 = 3(1) + 1
4 = 3 + 1
4 = 4
Equation 2
-(1) + 2(-2) = 5
-1 + (-4) = 5
-5 = 5
Equation 2
4 = -(1) + 5
4 =
4 = 4
(1,-2) is not a solution to the system of linear equations.
(1,4) is a solution to the system of linear equations.

6. Solving Systems of Linear Equations by Graphing
Step 1: Graph each equation in the same coordinate plane. Step 2: Estimate the point of intersection. Step 3: Check the point from step 2 by substituting for x and y in each equation of the original system.

7. Example 2: Solving a System of Linear Equations by Graphing
a) Solve the system of Linear equations by graphing: y = -2x + 5 Equation 1 y = 4x – 1 Equation 2
2) The two lines intersect at point (1, 3). 1)

8. (1,3) is the solution to the system.
3) Check your work:
y = -2x Equation y = 4x – Equation 2
3 = -2(1) + 5
3 = 3
3 = 4(1) – 1
3 = 3
(1,3) is the solution to the system.

9. Example 2: Solving a System of Linear Equations by Graphing
b) Solve the system of Linear equations by graphing: 2x + y = 5 Equation 1 3x – 2y = 4 Equation 2 1)
2x + 0 = 5
2x = 5
x = 5/2 = 2 ½
2(0) + y = 5
y = 5
3x – 2(0) = 4
3x = 4
x = 4/3 = 1 1/3
3(0) – 2y = 4
-2y = 4
y = -2
2) The two lines intersect at point (2, 1).

10. (2,1) is the solution to the system.
3) Check your work:
2x + y = Equation x – 2y = Equation 2
2(2) + 1 = 5
5 = 5
3(2) – 2(1) = 4
6 – 2 = 4
4 = 4
(2,1) is the solution to the system.

11. You try! (3,1) is the solution.
Solve the system of linear equations by graphing. y = x – 2 Equation 1 y = -x + 4 Equation 2
The lines intersect at (3,1).
Remember to substitute those values back into your original equations. Did they work?
(3,1) is the solution.

12. Example 3: Solving Real-Life Problems
A roofing contractor buys 30 bundles of shingles and 4 rolls of roofing paper for $1040. In a second purchase (at the same prices), the contractor buys 8 bundles of shingles for $256. Find the price per bundle of shingles and the price per roll of roofing paper. 30∙𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑏𝑢𝑛𝑑𝑙𝑒+4∙𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑟𝑜𝑙𝑙=1040 8∙𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑏𝑢𝑛𝑑𝑙𝑒+0∙𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑟𝑜𝑙𝑙=256
Let x be the price (in dollars) per bundle.
Let y be the price (in dollars) per roll.
𝑆𝑦𝑠𝑡𝑒𝑚 : 30𝑥+4𝑦=1040
8𝑥=256

13. 𝑆𝑦𝑠𝑡𝑒𝑚 : 30𝑥+4𝑦=1040 Equation 1 8𝑥=256 Equation 2
It looks as if the lines intersect at (32,20). This would mean that the price per bundle of shingles is $32 and the price per roll of roofing paper is $20.

14. Remember you can check your solution by substituting the x and y values into the original equations!!
Equation 1
30x + 4y = 1040
30(32) + 4(20) = 1040
1040 = 1040
Equation 2
8x = 256
8(32) = 256
256 = 256

15. You try!
You have a total of 18 math and science exercises for homework. You have 6 more math exercises than science exercises. How many exercises do you have in each subject?
(# of math exercises) + (# of science exercises) =18
(# of math exercises) – 6 = # of sciences exercises
Let x be the number of math exercises.
Let y be the number of science exercises.
𝑆𝑦𝑠𝑡𝑒𝑚 : 𝑥+𝑦=18
𝑥−6=𝑦

16. Check your solution! 𝑆𝑦𝑠𝑡𝑒𝑚 : 𝑥+𝑦=18 𝑥−6=𝑦
𝑆𝑦𝑠𝑡𝑒𝑚 : 𝑥+𝑦=18 𝑥−6=𝑦
The lines intersect at (12, 6). This means that you have 12 math exercises and 6 science exercises.
Check your solution!