Question 19·Easy·Systems of Two Linear Equations in Two Variables
Consider the system of equations.
The solution to the system is . What is the value of ?
(Express the answer as an integer)
For systems of two linear equations, quickly check if adding or subtracting the equations will eliminate a variable; if so, use elimination to get a simple one-variable equation, solve it, then substitute back into either original equation to find the other variable. Work carefully with negative signs and, if time permits, plug your solution into both original equations to confirm it satisfies both.
Hints
Look for an easy elimination
Notice that one equation has and the other has . What happens if you add the two equations together?
Solve for one variable first
After you eliminate by adding the equations, you should get an equation with only . Solve that to find .
Substitute back carefully
Once you know , plug it into either original equation and solve step by step for , paying close attention to negative signs.
Desmos Guide
Enter the equations
In Desmos, type 3x + y = 11 on one line and x - y = 1 on another line so both lines are graphed.
Find the intersection point
Look for the point where the two lines intersect; Desmos will show its coordinates when you tap or click on the intersection.
Read off the requested value
From the intersection point displayed by Desmos, note the second coordinate (the y-value); that is the value of that solves the system.
Step-by-step Explanation
Eliminate one variable
We have the system:
Add the two equations so that cancels:
This simplifies to:
Solve for :
Substitute to get an equation in y
Now substitute into one of the original equations, for example :
Rearrange this to isolate on one side:
so
Now you just need to solve this last simple equation for .
Solve for y and state the answer
From
multiply both sides by (or divide by ):
So, the value of in the solution is .