Question

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Explanation:

Step1: Verify table for $y=2x$

Substitute each $x$ into $y=2x$:

• $x=-4$: $y=2(-4)=-8$
• $x=-3$: $y=2(-3)=-6$
• $x=-2$: $y=2(-2)=-4$
• $x=-1$: $y=2(-1)=-2$
• $x=0$: $y=2(0)=0$
• $x=1$: $y=2(1)=2$
• $x=2$: $y=2(2)=4$
• $x=3$: $y=2(3)=6$
• $x=4$: $y=2(4)=8$

Step2: Verify table for $y=x-1$

Substitute each $x$ into $y=x-1$:

• $x=-4$: $y=-4-1=-5$
• $x=-3$: $y=-3-1=-4$
• $x=-2$: $y=-2-1=-3$
• $x=-1$: $y=-1-1=-2$
• $x=0$: $y=0-1=-1$
• $x=1$: $y=1-1=0$
• $x=2$: $y=2-1=1$
• $x=3$: $y=3-1=2$
• $x=4$: $y=4-1=3$

Step3: Find system solution

Find $x$ where $2x = x-1$:
$2x - x = -1$
$x=-1$
Substitute $x=-1$ into $y=2x$: $y=2(-1)=-2$

Step4: Graph explanation for part b

Plot points from the table for each line:

• For $y=2x$: $(-4,-8), (-3,-6), (-2,-4), (-1,-2), (0,0), (1,2), (2,4), (3,6), (4,8)$
• For $y=x-1$: $(-4,-5), (-3,-4), (-2,-3), (-1,-2), (0,-1), (1,0), (2,1), (3,2), (4,3)$

Draw straight lines through each set of points; they intersect at $(-1,-2)$.

Step5: Reflect question explanation

The system's solution is the only point that satisfies both individual equations, lying on both lines.

Answer:

Part a:

Completed table:

$x$	-4	-3	-2	-1	0	1	2	3	4
$y=x-1$	-5	-4	-3	-2	-1	0	1	2	3

Solution to the system: $(-1,-2)$. This is the only pair where both equations have the same $x$ and $y$ value, so it satisfies both equations.

Part b:

When graphing $y=2x$ (a line through the origin with slope 2) and $y=x-1$ (a line with y-intercept -1 and slope 1), the lines intersect at $(-1,-2)$, confirming the solution is reasonable.

Reflect:

The solution to the system is the single ordered pair that is a solution to both individual equations $y=2x$ and $y=x-1$; it is the intersection point of the two lines representing the individual equations.