Question

becca graphs the equations $y = -3(x - 1)$ and $y = x - 5$ to solve the equation $-3(x - 1) = x - 5$. her graph is shown below.
what are the solution(s) of $-3(x - 1) = x - 5$
-5 and 3

Explanation:

Step1: Recall intersection of graphs

To solve \(-3(x - 1)=x - 5\) using the graphs of \(y=-3(x - 1)\) and \(y=x - 5\), we find the \(x\)-coordinate of the intersection point of the two lines.

Step2: Solve the equation algebraically (to verify)

Start with \(-3(x - 1)=x - 5\).
Expand left side: \(-3x+3=x - 5\).
Subtract \(x\) from both sides: \(-4x + 3=-5\).
Subtract 3 from both sides: \(-4x=-8\).
Divide by \(-4\): \(x = 2\). Wait, but let's check the graph. Wait, maybe I made a mistake. Wait, let's re - solve:
\(-3(x - 1)=x - 5\)
\(-3x+3=x - 5\)
\(3 + 5=x + 3x\)
\(8 = 4x\)
\(x = 2\). Then find \(y\): \(y=2-5=-3\). So the intersection point is \((2,-3)\). But the options given? Wait, maybe the original problem's options were mis - presented. Wait, no, let's re - examine. Wait, the user's question: "What are the solution(s) of \(-3(x - 1)=x - 5\)?" with an option "-5 and 3" which is wrong. Wait, no, maybe I misread the equations. Wait, the first equation is \(y=-3(x - 1)\), the second is \(y=x - 5\). To find the solution of \(-3(x - 1)=x - 5\), we set the two \(y\)s equal, so the \(x\)-value of the intersection point is the solution. Let's solve the equation:
\(-3(x - 1)=x - 5\)
\(-3x + 3=x - 5\)
\(-3x-x=-5 - 3\)
\(-4x=-8\)
\(x = 2\). Then \(y=2-5=-3\). So the solution is \(x = 2\). But the option given is wrong. Wait, maybe the equations were different. Wait, if the first equation was \(y=-3(x + 1)\), then:
\(-3(x + 1)=x - 5\)
\(-3x-3=x - 5\)
\(-3x-x=-5 + 3\)
\(-4x=-2\)
\(x=\frac{1}{2}\), no. Wait, maybe the user made a typo. But according to the correct solution of \(-3(x - 1)=x - 5\), the solution is \(x = 2\). But if we consider the intersection of the two lines, the \(x\)-coordinate is the solution.

Wait, maybe the original problem had a different set of equations. Wait, let's check the graph again. The first line (steep negative slope) has a \(y\)-intercept at \(y = 3\) (since \(y=-3(x - 1)=-3x + 3\)), and the second line (positive slope) has a \(y\)-intercept at \(y=-5\) (since \(y=x - 5\)). The intersection point: let's find where they cross. Let's set \(-3x + 3=x - 5\), so \(4x=8\), \(x = 2\), \(y=-3\). So the solution is \(x = 2\). But the option given is wrong. However, if we assume that there was a mistake in the problem and the equations were \(y=-3(x + 1)\) and \(y=x - 5\), no. Alternatively, maybe the question was to solve \(-3(x - 1)=y\) and \(y=x - 5\) for \(x\) and \(y\), but the solution to \(-3(x - 1)=x - 5\) is \(x = 2\).

But since the user provided an option "-5 and 3" which is incorrect, but according to the correct mathematical solution:

Answer:

The solution of the equation \(-3(x - 1)=x - 5\) is \(x = 2\). (Note: The option "-5 and 3" is incorrect. The correct solution is obtained by finding the \(x\)-coordinate of the intersection point of the lines \(y=-3(x - 1)\) and \(y=x - 5\), which is \(x = 2\) after solving the equation algebraically as shown above.)