The system of equations is
For each real number , which of the following points lies on the graph of each equation in the -plane?
A parameterized point uses a letter to describe a point that moves. If a system asks for a point on both graphs for every value of that letter, graph the equations in Desmos, then check whether one equation is a scaled version of the other. Choose one coordinate to equal the letter and solve for its partner. Watch the sign when dividing by a negative coefficient.
Hints
- Hint 1
A system requires both equations to hold at once. Multiplying every term of an equation by the same nonzero number keeps its solutions, so check whether the two equations describe the same line.
- Hint 2
An ordered pair puts first and second. To get a point for every , choose the first coordinate to be . What does the shared equation become when ?
- Hint 3
To isolate means to get it alone on one side. After putting , subtract the term containing , then divide by the coefficient of . Keep the minus sign on that coefficient.
Step-by-step
Find the shared line, then set x equal to t
Step 1Graph both equations
Type and on separate Desmos lines. Their graphs, the points satisfying each equation, overlap rather than crossing at one point. Check that the equations really give the same line before choosing a point.
- Step 2
Confirm the lines are identical
Divide the first equation by :
Multiply this whole equation by :
That matches the second equation. Scaling the entire equation, including its constant, by a nonzero number keeps every solution point. So any point on lies on both graphs.
- Step 3
Give t a coordinate
An ordered pair lists first. Since can be any real number, set in the shared equation:
- Step 4
Find the matching y-coordinate
Subtract from both sides:
Divide by , keeping its negative sign:
So for every real , the point lies on the graph of each equation. Choice C.