The graphs of the equations in the given system intersect at the point in the -plane. What is the value of ?
For systems where one equation directly gives a value for or , immediately substitute that value into the other equation instead of trying to graph in your head. Work step by step, especially with negatives: square the number first, then handle multiplications, and only then add or subtract. This minimizes sign errors and lets you reach the correct value quickly without unnecessary algebra.
Hints
Use the definition of intersection
An intersection point must satisfy both equations in the system at the same time. One equation already tells you the value of .
Plug x into the second equation
You know that . Put this value into the equation to express in terms of numbers only.
Be careful with negatives
When you simplify , handle the square and the product with negatives separately. What is ? What is times ?
Combine your results
After you find each part of the expression, add them together to get the final value of .
Desmos Guide
Enter the two equations
Type y = x^2 - 2x into one expression line and x = -3 into another line. This will graph a parabola and a vertical line.
Find the intersection point
Tap or click on the point where the vertical line crosses the parabola. Desmos will display the coordinates of this point; note the -value shown there.
Step-by-step Explanation
Understand what “intersection” means
For two equations in a system, an intersection point is a pair of values that makes both equations true at the same time. Here, the first equation already tells you the -coordinate: .
Substitute the given x-value into the y-equation
The second equation is . Since the intersection must have , replace with :
This gives an expression you can simplify to find .
Simplify carefully using order of operations
First square :
Then multiply and :
So the expression for becomes:
Add to find the y-value
Now add the two results:
So, the -value where the graphs intersect is 15, which corresponds to choice C.