Question

Courtney believes that, given a system of two linear equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solution. Test Courtney's conjecture by completing the parts below.

Part A: Verify that the system of equations 3x - 7y = 26 and 5x + 2y = 16 has a solution of (4, -2) by substituting the x- and y-coordinates of the point into both equations and simplifying.
Part B: Multiply both sides of the equation 3x - 7y = 26 by 2 and then add the result to the equation 5x + 2y = 16. Replace the equation 5x + 2y = 16 with the sum that you just found to create a new system of equations. Show your work.
Part C: Show whether or not (4, -2) is a solution to the new system of equations by substituting the x- and y-coordinates of the point into both equations and simplifying. Is Courtney's conjecture correct based on this example?

Answer 1

Upon testing, substituting (4, -2) into the original system of equations proves true, thereby verifying it as a solution. After modifying one of the equations as proposed, the system still has the same solution with (4, -2) satisfying both equations. Hence, Courtney's conjecture is correct based on this example.

Step-by-step explanation:

Part A: Verify the Solution

To verify that (4, -2) is a solution for the system of equations 3x - 7y = 26 and 5x + 2y = 16, we substitute x with 4 and y with -2.

• For the first equation, 3(4) - 7(-2) = 12 + 14 = 26, which matches the equation.
• For the second equation, 5(4) + 2(-2) = 20 - 4 = 16, which also matches the equation.

This confirms that (4, -2) is indeed a solution to both equations.

Part B: Create a New System

Multiplying the first equation 3x - 7y = 26 by 2 gives us 6x - 14y = 52. Adding this result to the second equation, we get:

• 6x - 14y + 5x + 2y = 52 + 16
• 11x - 12y = 68

Part C: Testing the Solution

We test if (4, -2) is a solution for the new system which includes the equations 3x - 7y = 26 and 11x - 12y = 68:

• The original first equation is unchanged, and we have already verified that (4, -2) is a solution.
• For the new equation, 11(4) - 12(-2) = 44 + 24 = 68, which means (4, -2) is a solution to this equation as well.

Thus, Courtney's conjecture is correct based on this example, as the solution set remains the same after the alteration of the equations.