Systems of equations and the substitution method

Satisfying equations Suppose that x + y = 3. How many pairs of values make this equation true? x = 1 and y = 2, x = 3 and y = 0, x = –2 and y = 5… and so on. Equations that contain two unknowns have an infinite number of solution pairs. Suppose we have two of these sorts of linear equations in two variables, for example, x + y = 3 and y – x = 1. Teacher notes Ask students to suggest other pairs of values that would solve this equation. Praise correct answers involving negative or fractional solutions. Verify that all of the suggested solutions lie on the line x + y = 3. How many pairs of values satisfy both equations? There is just one solution pair: x = 1 and y = 2.

Systems of equations When we have two or more equations containing the same unknowns we call them a system of equations. We know how to solve an equation in one variable; we use inverse operations. Suppose that y = 2x – 3 and x + 2y = 14. It is difficult to see the solution to these two equations. Can you think of a method to help us find it? One method we can use to easily solve a system of two equations is by substituting one equation into the other. Teacher notes This slide is the first of two slides taking students through how the substitution method works for the two given equations. Mathematical Practices 7) Look for and make use of structure. Students should be able to see that, in the first equation, we are given a clear definition of the value of y in terms of x: “y = 2x – 3”. They should hopefully use the statement “We know how to solve an equation in one variable” to make the connection that they can substitute the first equation into the second in order to get one equation in one variable. This gives us one equation in one variable, which we know how to solve, then we can easily find the second variable.

The substitution method We can solve y = 2x – 3 and x + 2y = 14 by substitution. Label the starting equations: y = 2x – 3 A x + 2y = 14 B Equation gives y in terms of x. A Substitute equation into equation to give one equation in x: A B x + 2(2x – 3) = 14 Teacher notes This slide is the first of two slides taking students through how the substitution method works for one example. Image credit: © Seregam, Shutterstock.com 2012 distributive property: x + 4x – 6 = 14 simplify: 5x – 6 = 14 5x = 20 add 9 to both sides: x = 4 divide both sides by 8:

The substitution method We now need to find the value of y when x = 4. Substitute this value into one of the original equations: y = 2x – 3. y = 2(4) – 3 y = 5 Our solution is: x = 4, y = 5. Why do you think it is important to substitute into the other equation to check your solution? Check these answers by substituting x = 4 and y = 5 into the other equation: x + 2y = 14 Teacher notes Emphasize to students that if they use y = 2x – 3 to find the value of y, then also check their solution using the same equation, this is not very accurate. If they have made a mistake at any point, their solution still might satisfy the first equation but might NOT satisfy the second equation. By using one equation to find the second variable, then checking with the other equation, we can be sure that our solution satisfies both. Mathematical Practices 6) Attend to precision. Students should realize the importance of checking their answers when working with systems of equations. If they have made a mistake at some point in their calculations, the pair of values they find may well satisfy one equation but not the other. Image credit: © BruceParrott, Shutterstock.com 2012 (4) + 2(5) = 14 This is true, so we know that our solutions are correct. 4 + 10 = 14

The substitution method How could the following system of equations be solved using substitution? 3x – y = 9 A 8x + 5y = 1 B One of the equations needs to be arranged in the form x = … or y = … before it can be substituted into the other equation. Rearrange equation : A Teacher notes This slide is designed to make students realize themselves that sometimes one equation needs to be rearranged to get it in the form y = mx + b. Mathematical Practices 7) Look for and make use of structure. Should be able to look at the two equations and see that the substitution method can still be used if the first equation is rearranged into the form y = mx + b. 3x – y = 9 add y : 3x = 9 + y subtract 9 : 3x – 9 = y y = 3x – 9

The substitution method Now we have: y = 3x – 9 A 8x + 5y = 1 B We can now substitute y = 3x – 9 into equation . B 8x + 5(3x – 9) = 1 8x + 15x – 45 = 1 distributive property: 23x – 45 = 1 simplify: 23x = 46 add 45: divide by 23: x = 2 Photo credit: © Seregam, Shutterstock.com 2012 Substitute x = 2 into equation to find the value of y. A 3(2) – y = 9 6 – y = 9 simplify: –y = 3 subtract 6: y = –3 multiply by –1:

The substitution method Our equations: y = 3x – 9 A 8x + 5y = 1 B Check the solutions x = 2 and y = –3 by substituting them into equation . B 8(2) + 5(–3) = 1 16 – 15 = 1 This is true, so we know that our solutions are correct.