Question 110·Medium·Systems of Two Linear Equations in Two Variables
In the solution to the system of equations above, what is the value of ?
For linear systems on the SAT, quickly choose the method (substitution or elimination) that makes the algebra shortest—usually by solving the equation where a variable has coefficient 1 or -1. Carefully substitute to find the needed variable, but always double-check what the question is actually asking for (like , , or ) so you perform the final calculation correctly instead of stopping at or alone.
Hints
Choose a method to solve the system
Decide whether substitution or elimination is easier. Which equation makes it easiest to isolate or ?
Use the equation that is easy to rearrange
Try solving the second equation, , for and then substitute that expression into the first equation.
Focus on what the question actually asks
After you find the value of , you are not done yet. You still need to use that value to compute .
Desmos Guide
Graph the system in Desmos
In Desmos, enter the two equations exactly as they are, each on its own line:
2x + 3y = 74x - y = 5Desmos will graph both lines and show their intersection point.
Find the x-value of the intersection
Tap or click on the point where the two lines intersect. Note the -coordinate of this point; that is the value of that solves the system.
Use Desmos to compute 7x
In a new expression line, type 7 * (x-value) using the -coordinate you found from the intersection. The resulting output is the value of ; compare this number to the answer choices.
Step-by-step Explanation
Isolate one variable using the simpler equation
Look at the second equation, , because it is easy to solve for .
From :
Now you have written in terms of .
Substitute into the first equation and solve for x
Substitute into the first equation :
Distribute the 3:
Combine like terms:
Add 15 to both sides:
Divide both sides by 14:
So the solution’s -value is .
Write an expression for 7x using the x-value
The question does not ask for ; it asks for .
Using , write:
Now simplify this product.
Simplify 7x and choose the matching option
Compute the product:
So , which matches answer choice B) 11.