Question 72·Hard·Systems of Two Linear Equations in Two Variables
Solve the system of equations.
The solution to the system is . What is the value of ?
When a system-of-equations question asks for a specific linear combination like 4x + 6y, do not automatically solve for x and y. Instead, look for ways to add, subtract, or multiply the original equations so that their left-hand sides combine to the exact expression you need; then you can add or subtract the right-hand sides in the same way to get the value directly, which is usually faster and less error-prone than solving the full system.
Hints
Avoid solving for x and y directly
Instead of finding x and y separately, focus on how to combine the two equations to get the expression 4x + 6y.
Break 4x + 6y into simpler pieces
Try writing 4x + 6y as a sum of two smaller expressions involving x and y, such as (x + 2y) plus another combination that looks similar to one of the equations.
Use addition and subtraction of equations
See what you get if you subtract the second equation from the first, and what you get if you multiply the second equation by 2. Do any of those left-hand sides look like the pieces you need to build 4x + 6y?
Desmos Guide
Graph the two lines
In Desmos, enter each equation exactly as given on its own line: (5/2)x + 4y = 31 and (3/2)x + 2y = 19. Desmos will draw both lines.
Find the intersection point
Click or tap where the two lines intersect. Desmos will show the intersection coordinates (x, y); these are the solution to the system.
Compute 4x + 6y using the intersection
In a new expression line, type 4*(x-coordinate) + 6*(y-coordinate) using the numbers from the intersection point. The resulting value shown by Desmos is the value of 4x + 6y.
Step-by-step Explanation
Notice how 4x + 6y relates to the given equations
You do not need the individual values of x and y if you can directly form 4x + 6y from the equations.
Observe that:
So the plan is to create equations for x + 2y and for 3x + 4y from the given system, then add them.
Create an equation for x + 2y
Subtract the second equation from the first:
Given:
Subtract the second from the first (left side minus left side, right side minus right side):
This simplifies to:
So:
Create an equation for 3x + 4y
Now use the second equation again:
Multiply the entire equation by 2 so that the fractions clear:
This gives:
Add the two new equations to get 4x + 6y
Now add the equations you found:
The left side is exactly 4x + 6y, and the right side is 50, so:
Therefore, the value of is 50.