Question 81·Hard·Systems of Two Linear Equations in Two Variables
The graph shows two distinct lines, and , on the -plane. Each line passes through two labeled points.
Which choice gives the exact coordinates of the intersection point of lines and ?
For systems shown as two intersecting lines, the solution is their intersection. When the graph gives two points on each line, first write each line’s equation (compute slope, use point-slope, then convert to standard form). Then solve the system using elimination to avoid fraction-heavy substitution, and only at the end convert the results into an ordered pair for the answer choice.
Hints
Turn each line into an equation
Use the two labeled points on each line to find its slope, then write an equation (point-slope form works well).
Put both equations in standard form
Rewrite each line as so it’s easy to use elimination.
Use elimination to find the intersection
Multiply equations so either the -terms or -terms match, subtract to eliminate a variable, and then substitute back to get the other coordinate.
Desmos Guide
Enter the two line equations
In Desmos, enter the equations for the lines: and .
Find the intersection point
Click on the point where the two lines cross to display its coordinates.
Match the displayed coordinates to the choices
Use the exact coordinates shown for the intersection to choose the matching ordered pair.
Step-by-step Explanation
Write an equation for line
From the graph, line passes through and .
Using point-slope form with :
Multiply by and rearrange:
Write an equation for line
From the graph, line passes through and .
Using point-slope form with :
Multiply by and rearrange:
Eliminate one variable to find
Eliminate by making the -coefficients match.
Multiply by and by :
Subtract the first equation from the second:
Substitute to find and form the intersection point
From , we have .
Substitute into :
Therefore, the intersection point is .