A dependent system has two equations that draw the same line. A question asking for a point for every value of a letter is a cue to check whether the equations are multiples of each other. Graph both in Desmos, then solve the shared equation for and use a first coordinate from the choices. Changing one coordinate without changing the other to match is the trap.
Hints
- Hint 1
A system asks for points that satisfy both equations. Check whether multiplying every term of the first equation, including the number on the right, produces the second. What would that mean for their graphs?
- Hint 2
An ordered pair lists first and second. Solve the shared line for so you know which second coordinate must go with any first coordinate.
- Hint 3
Two choices use the first coordinate . Put that expression in for in your formula. What must the second coordinate be for every value of ?
Step-by-step
Find the shared line's point rule
Step 1Graph both equations
Type both equations into Desmos. Their graphs overlap.
- Step 2
Check whether the equations share a line
The -coefficients give . Multiplying the entire first equation by , including its right side, gives: . So every point on the first line lies on the second too.
- Step 3
Find the second coordinate from the first
Because the equations share a line, use to find , the second coordinate of a point . Subtract from both sides: . Divide both sides by :
- Step 4
Match a point that works for every value
Two choices have first coordinate . Put that into the rule. Substitute: . Distribute : . Combine the constants: . For every , the second coordinate must be the one the line's equation gives for the first. So lies on both graphs for any real . Choice A.