Chapter 4, Problem 39SGR

To determine

To Write:

Write an equation in slope-intercept form for the line that passes through the given point and is parallel to the graph of each equation.

Answer to Problem 39SGR

The equation of line in slope-intercept form that passes through point (2,5) and is parallel to the line $y=x-3$

  $y=x+3$

Explanation of Solution

Given Information:

Line passes through point (2,5) and is parallel to the line $y=x-3$

Formula Used:

• Two lines are parallel if they have same slope.
• The slope-intercept form of a line is :

  $\begin{array}{l}y=mx+b\\ where\\ m=slope\\ b=y-\mathrm{int}ercept\end{array}$

• The point-slope equation of a line is :

  $\begin{array}{l}y-y_1=m(x-x_1)\\ where\\ m=slope\\ (x_1,y_1)=po\mathrm{int}\_through\_which\_line\_passes\end{array}$

Calculation:

Since the required line passes through point (2,5) and is parallel to the line $y=x-3$ .

1. Finding the slope of required line :
• Comparing the equation $y=x-3$ with slope-intercept form of a line

$\begin{array}{l}y=mx+b\\ where\\ m=slope\\ b=y-\mathrm{int}ercept\\ y=x-3\\ \Rightarrow m=1\\ \&\\ b=y-\mathrm{int}ercept=-3\end{array}$

• Since the slope of parallel lines is equal .
• So, the slope of required line is 1.
2. Write the equation :
• From the point-slope form of a line , since the required line passes through (2,5) .
• Equation of required line is :

  $\begin{array}{l}y-y_1=m(x-x_1)\mathrm{.....................}[point-slopeform]\\ m=slope\\ (x_1,y_1)=po\mathrm{ints}\_through\_which\_line\_passes\\ \Rightarrow y-5=1(x-2)\mathrm{.....................}[m=1;(x_1,y_1)=(2,5)]\\ \Rightarrow y-5=x-2\\ \Rightarrow y-5-x=x-2-x\mathrm{..............}[subtracting"x"each\_side]\\ \Rightarrow y-x-5=-2\\ \Rightarrow y-x-5+5=-2+\mathrm{5.............}[add"5"each\_side]\\ \Rightarrow y-x=3\\ or\\ y=x+3\end{array}$

So, the equation of line in slope-intercept form that passes through point (2,5) and is parallel to the line $y=x-3$

  $y=x+3$