Question

practise

1. one linear system is shown in the table of values and another in the graph. do the two systems have the same solution? justify your answer.

Explanation:

Step1: Find table system's solution

We need to find the $x$-value where $Y_1 = Y_2$. Looking at the table:
When $x=2$, $Y_1=1$, $Y_2=7$; when $x=3$, $Y_1=0$, $Y_2=9$; we can derive the equations first.
For $Y_1$: slope $m_1=\frac{0-1}{3-2}=-1$, using point $(2,1)$: $Y_1 = -x + 3$.
For $Y_2$: slope $m_2=\frac{9-7}{3-2}=2$, using point $(2,7)$: $Y_2 = 2x + 3$.
Set $-x + 3 = 2x + 3$
$-3x = 0$
$x=0$, substitute $x=0$: $Y_1=3$, $Y_2=3$. So solution is $(0, 3)$.

Step2: Find graph system's solution

The solution of a linear system on a graph is the intersection point of the two lines. From the graph, the lines intersect at $(1, 1)$.

Step3: Compare the two solutions

The table system's solution is $(0,3)$, the graph system's solution is $(1,1)$. These are not the same point.

Answer:

No, the two systems do not have the same solution. The table-based linear system has a solution of $(0, 3)$, while the graph-based linear system has a solution of $(1, 1)$, which are distinct points.