5.1 Solving Systems of Linear Equations by Graphing

System of Linear Equations A set of two or more linear equations in the same variables Example: x + y = 7 Equation 1 2x – 3y = -11 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

Example 1: Checking Solutions Tell whether the ordered pair is a solution of the system of linear equations. (2,5); x + y = 7 Equation 1 2x – 3y = -11 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.

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 = -2 + 4 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.

You try! Tell whether the ordered pair is a solution to the system of linear equations. (1, -2); 2x + y = 0 2) (1,4); y = 3x + 1 -x + 2y = 5 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 = -1 + 5 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.

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.

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)

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

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).

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

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.

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

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.

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

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

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!