Chapter 1, Problem 35RE

To determine

To find: The Equation of the line

Answer to Problem 35RE

The equation of the line in the general form is $x-y=7$

Explanation of Solution

Given:

Perpendicular to the line $x+y=2$ a

  $\left(4,-3\right)$

Formula Used:

The slope-intercept form $y=mx+b$

Calculation:

For finding the slope, convert the line $x+y=2$ into its slope-intercept form, $y=mx+b$ .

Subtract $x$ from both sides of the equation to isolate $y$ .

  $x+y-x=2-x$

  $y=-x+2$

Comparing $y=-x+2$ with $y=mx+b$ , we get the value of $m$ as $-1$ .

Since the two lines are perpendicular, the product of their slopes must be $-1$ .

Let $m_1$ be the slope of the line perpendicular to $x+y=2$ .

  $m{m}_1=-1$

  $-1m_1=-1$

Divide by $-1$ on both sides of the equation

  $\frac{-1m_1}{-1}=\frac{-1}{-1}$

  $m_1=1$

Thus, the slope $m_1$ is $1$ .

Since the point $\left(4,-3\right)$ lies on the line withslope 1, use the point-slope form to obtain the equation of the line.

  $y-y_1=m\left(x-x_1\right)$

  $y-\left(-3\right)=1\left(x-4\right)$

Remove the parentheses using the distributive property of multiplication.

  $y+3=x-4$

Subtract 3 from both sides of the equation.

  $y+3-3=x-4-3$

  $y=x-7$

Therefore, the slope-intercept form of the equation is $y=x-7$.

In order to get the equation of the line in the general form, simplify $y=x-7$.

Subtract $x$ from both sides of the equation.

  $y-x=x-7-x$

  $y-x=-7$

Multiply by $-1$ on both sides of the equation.

  $\left(-1\right)y-\left(-1\right)x=\left(-1\right)\left(-7\right)$

  $x-y=7$

Conclusion:

Therefore, the equation of the line in the general form is $x-y=7$