If is the solution to the system of equations above, what is the value of ?
In a linear system, the solution makes both equations true. Graph the equations in Desmos and click their intersection, then add the coordinates if the question asks for . When coefficients swap places, adding the equations may offer a faster shortcut. The right-side total alone isn't necessarily the sum you need.
Hints
- Hint 1
An intersection is a point on both lines. Type each equation into Desmos as written, then click where the lines cross. That point gives values of and that satisfy both equations.
- Hint 2
In the ordered pair , the first coordinate is and the second is . Once you read both coordinates, what must you do to find ?
Step-by-step
Approach 1: Read the intersection in Desmos
Step 1Find the pair that satisfies both equations
Type and on separate lines. Click their intersection, the point on both graphs. Desmos shows , so this pair satisfies both equations.
- Step 2
Add the coordinates
The first coordinate is and the second is . Type ; Desmos prints . Use the solution pair to build the expression the question asks for. Here, the value of is . Choice C.
Approach 2: Build the sum directly
Step 1Add the two equations
No Desmos needed. The swapped coefficients let you build the requested sum by adding. Add the left sides and the right sides, because equal amounts added to equal amounts stay equal:
- Step 2
Combine matching terms
Combine the -terms, the -terms, and the numbers on the right:
The is not yet ; each variable still has a coefficient of .
- Step 3
Show the requested sum
Factor from the left so the you're looking for appears together:
- Step 4
Find the value of the sum
Divide both sides by :
So the value of is . Choice C.