Question 129·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 ?
(Express the answer as an integer)
For systems of two linear equations on the SAT, elimination is usually the fastest and most reliable method, especially when the question asks for a specific expression like instead of the individual variables. Aim to eliminate one variable by multiplying the equations so that one pair of coefficients becomes opposites, then add or subtract the equations. Pay attention to the expression the question wants: if you directly obtain an equation for that expression (like ), you can stop there without fully solving for and , saving time and reducing opportunities for arithmetic mistakes.
Hints
Think about what the question is asking for
You do not actually need the individual values of and . The question only asks for . How could you combine the two equations to get an equation involving directly?
Eliminate one variable
Look at the coefficients of in the two equations. What number could you multiply each equation by so that the -terms become opposites and cancel when added?
Combine after scaling
After you multiply the first equation by 5 and the second by 4, add the new equations together. Focus on what the -terms become when you add them, and write the resulting equation for .
Desmos Guide
Graph the system
In Desmos, enter the two equations on separate lines: 7x-4y=6 and 3x+5y=44. You will see two straight lines on the graph.
Find the intersection point
Click on the point where the two lines intersect. Desmos will display the coordinates of this point as ; this is the solution to the system.
Calculate 47x from the intersection
In a new expression line, type 47* followed by the x-coordinate of the intersection point (you can type the number or click the x-value to paste it). The value that Desmos shows for this expression is the value of for the solution.
Step-by-step Explanation
Choose elimination to target 47x
Notice the system:
The question asks for , not for or separately. If we eliminate , the -terms will combine. Our goal is to combine the equations so that the coefficient of becomes 47.
Make the y-coefficients opposites
The -coefficients are and . Their least common multiple is 20.
- Multiply the first equation by 5:
- Multiply the second equation by 4:
Now the -terms are and , which will cancel when we add the equations.
Add the two new equations
Add the two scaled equations term by term:
The -terms disappear, leaving an equation involving only and the expression that we care about.
Compute 47x and answer
Add the numbers on the right side:
The question asks for , so the value we need to report is 206.