Question 1·Easy·Nonlinear Equations in One Variable; Systems in Two Variables
The graph of a system consisting of an absolute value function and a linear function is shown.
What is the solution to this system of two equations?
For SAT questions that show the graph of a system, do not waste time trying to solve algebraically unless necessary; instead, immediately look for the intersection point of the graphs. Carefully read the - and -coordinates using the grid, watch out for sign errors or swapped coordinates, and then select the answer choice that matches that ordered pair exactly.
Hints
What does the solution represent on a graph?
Think about how you can tell, just by looking at a graph, which points satisfy both equations in a system at the same time.
Focus on the intersection
Find the point where the straight line and the V‑shaped absolute value graph cross. That point is the solution to the system.
Read the ordered pair carefully
Once you see the intersection, use the grid: read the -value (left/right) and the -value (up/down). Be careful about the sign (positive vs. negative) and make sure you don’t swap and .
Desmos Guide
Translate each graph into an equation (optional)
From the SAT graph, read off clear points on the absolute value graph (the V-shape) and on the line. Use these points to write approximate equations, for example using slope for the line and the vertex plus another point for the absolute value. (Any correct equations that match the graph are fine.)
Enter the equations in Desmos
Type your two equations into Desmos, one per line. You should see a V‑shaped graph for the absolute value function and a straight line that look similar to the SAT graph.
Use Desmos to find the intersection
Tap or click on the point where the two graphs cross. Desmos will display the coordinates of that intersection; those coordinates give you the solution to the system. Match that ordered pair to one of the answer choices.
Step-by-step Explanation
Understand what “solution to a system” means
A solution to a system of two equations is a pair that makes both equations true at the same time.
On a graph, that means the solution is the point where the two graphs intersect (cross). So we just need to find the intersection of the absolute value graph and the line.
Locate the intersection point on the graph
Look at the graph and find the one point where the V-shaped absolute value graph and the straight line cross.
Visually identify that intersection point, then check the grid lines around it to see exactly which vertical and horizontal grid lines it lies on.
Read the coordinates of the intersection
To write the coordinates of the intersection:
- First read the -coordinate by seeing which vertical grid line the point is above (how far left or right of the origin it is).
- Then read the -coordinate by seeing which horizontal grid line the point is on (how far up or down from the origin it is).
On this graph, the intersection is 1 unit to the right of the origin and 5 units up, so its coordinates are .
Therefore, the solution to this system of two equations is .