Question

the point where the graphs of two equations intersect has a y-coordinate of 2. one equation is $y = -3x + 5$. determine the other equation if its graph has a slope of 1.

Explanation:

Step1: Find the x - coordinate of the intersection point

We know that at the intersection point, the \(y\) - coordinate is 2 and the point lies on the line \(y=-3x + 5\). Substitute \(y = 2\) into the equation \(y=-3x+5\):
\[2=-3x + 5\]
Subtract 5 from both sides:
\[2-5=-3x\]
\[-3=-3x\]
Divide both sides by - 3:
\[x = 1\]
So the intersection point is \((1,2)\).

Step2: Find the equation of the line with slope 1 passing through \((1,2)\)

The slope - intercept form of a line is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. We know that \(m = 1\) and the line passes through the point \((x,y)=(1,2)\). Substitute \(x = 1\), \(y = 2\) and \(m = 1\) into the equation \(y=mx + b\):
\[2=1\times1 + b\]
\[2=1 + b\]
Subtract 1 from both sides:
\[b=2 - 1=1\]

Since \(m = 1\) and \(b = 1\), the equation of the line is \(y=x + 1\).

Answer:

\(y=x + 1\)