Question

1. these two lines represent a system of equations. what is the y - coordinate of the point that makes both equations true?

a. 0
b. \\(\frac{1}{5}\\)
c. 3
d. 4

spiral review
problems 7–9: consider the equation \\(4x - 4 = 4x + \underline{}\\). what value or expression could you write in the blank so that the equation is true for:

1. no values of \\(x\\)?
2. all values of \\(x\\)?
3. one value of \\(x\\)?

Explanation:

Step1: Find intersection point (Q6)

The point satisfying both equations is where the lines cross. From the graph, this point is $(5, 3)$, so its $y$-coordinate is 3.

Step2: No solution condition (Q7)

For $4x - 4 = 4x + c$, subtract $4x$: $-4 = c$. For no solution, $-4
eq c$, e.g., $c=0$.

Step3: Infinite solutions condition (Q8)

For $4x - 4 = 4x + c$, subtract $4x$: $-4 = c$. So $c=-4$.

Step4: One solution condition (Q9)

For one solution, the equation must have a single $x$ term. Use a non-$x$ term that creates a solvable equation, e.g., $c=-x$. Then $4x -4 =4x -x \implies x=4$.

Answer:

1. C. 3
2. Any number except -4 (example: 0)
3. -4
4. Any linear expression with x (example: $-x$)