Question

Write the slope-intercept form of the equation of the line that passes through the two points. Graph the line. Label the points where the line crosses the axes. \((1,1),(-3,5)\)

Short answer

The line that passes through the points (1,1) and (-3,5) has the slope-intercept form \(y = -x + 2\). It intercepts the y-axis at the point (0,2) and x-axis at the point (2,0).

Step-by-step solution

Calculate the slope of the line

We begin by determining the slope of the line. The slope (m) formula given two points \((x1,y1)\) and \((x2,y2)\) is \[m = (y2-y1) / (x2-x1)\]. Thus, m where \((x1,y1)\) are (1,1) and \((x2,y2)\) are (-3,5) is \[m = (5-1) / (-3-1) = -1\].

Obtain the equation of the line

Next, we substitute the calculated slope (-1) and one of the provided points into the slope-intercept form \[y = mx + b\] in order to solve for the y-intercept (b). Using point (1,1) will give us the equation \(1 = -1*1 + b\). This simplifies to \(b = 2\). Hence the equation of the line is \(y = -x + 2\).

Determine the x and y intercepts

The x-intercept is the point at which the line crosses the x-axis (y=0). Thus, for the equation of the line \(y = -x + 2\), setting y=0, we obtain \(x = 2\). Thus the x-intercept is (2,0). Conversely, the y-intercept is point at which line crosses the y-axis (x=0). Therefore, setting x=0 in the equation of the line, we get \(y = 2\). This means the y-intercept is at the point (0,2).

Graph the line

Next, we draw a graph. We mark the y-intercept and x-intercept namely the points (0,2) and (2,0) respectively and proceed to draw a straight line through these points, ensuring also that it crosses the given points (1,1) and (-3,5). This results in a decreasing line with slope -1.