Question 35·Medium·Linear Equations in Two Variables
The system of equations is given.
What is the value of ?
For a linear system on the SAT, quickly choose between substitution and elimination by looking for easy coefficients: if a variable has coefficient 1 or -1, isolate that variable and substitute it into the other equation. Keep your algebra organized—write each step clearly when substituting and combining like terms, and be especially careful with negative signs and the final division step. If the question only asks for one variable, stop as soon as you find that variable rather than solving for both.
Hints
Pick the easier equation for substitution
Look at which equation makes it easiest to isolate one variable. One of them already has a coefficient of on .
Express one variable in terms of the other
Rearrange the simpler equation to solve for in terms of . Be careful with signs when you move terms across the equals sign.
Substitute and solve carefully
Plug your expression for into the other equation. Combine like terms, move constants to one side, and then divide to isolate .
Check for algebra errors
After you get a value for , mentally plug it back into one of the original equations to see if it approximately satisfies it (left side close to right side). If not, recheck your steps.
Desmos Guide
Graph both equations
In Desmos, enter 2x + 3y = 7 on one line and 5x - y = 4 on another line. Desmos will graph both lines on the coordinate plane.
Find the intersection point
Tap or click on the point where the two lines intersect. Desmos will display the coordinates of this intersection; the -coordinate of this point is the solution for in the system.
Step-by-step Explanation
Choose a method to solve the system
Since the second equation has a simple coefficient of on , it is easy to solve it for and then substitute into the first equation. This is called the substitution method.
Solve the second equation for
Start with the second equation:
Subtract from both sides:
Multiply both sides by to solve for :
Now you have written in terms of .
Substitute into the first equation and simplify
Use the expression in the first equation .
Substitute:
Distribute the 3:
Combine like terms:
Add 12 to both sides:
Now you just need to solve for by dividing.
Solve for and select the answer
From , divide both sides by 17:
This matches answer choice D) 19/17.