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Find the equation of the line parallel to 5 x = 7 + y and passing through the point ( −1 , −2 ) .

y = 5 x + 3

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Finding the equation of a line perpendicular to a given line passing through a given point

Find the equation of the line perpendicular to 5 x 3 y + 4 = 0 ( 4 , 1 ) .

The first step is to write the equation in slope-intercept form.

5 x 3 y + 4 = 0 −3 y = −5 x 4 y = 5 3 x + 4 3

We see that the slope is m = 5 3 . This means that the slope of the line perpendicular to the given line is the negative reciprocal, or 3 5 . Next, we use the point-slope formula with this new slope and the given point.

y 1 = 3 5 ( x ( −4 ) ) y 1 = 3 5 x 12 5 y = 3 5 x 12 5 + 5 5 y = 3 5 x 7 5
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Key concepts

  • We can solve linear equations in one variable in the form a x + b = 0 using standard algebraic properties. See [link] and [link] .
  • A rational expression is a quotient of two polynomials. We use the LCD to clear the fractions from an equation. See [link] and [link] .
  • All solutions to a rational equation should be verified within the original equation to avoid an undefined term, or zero in the denominator. See [link] and [link] .
  • Given two points, we can find the slope of a line using the slope formula. See [link] .
  • We can identify the slope and y -intercept of an equation in slope-intercept form. See [link] .
  • We can find the equation of a line given the slope and a point. See [link] .
  • We can also find the equation of a line given two points. Find the slope and use the point-slope formula. See [link] .
  • The standard form of a line has no fractions. See [link] .
  • Horizontal lines have a slope of zero and are defined as y = c , where c is a constant.
  • Vertical lines have an undefined slope (zero in the denominator), and are defined as x = c , where c is a constant. See [link] .
  • Parallel lines have the same slope and different y- intercepts. See [link] .
  • Perpendicular lines have slopes that are negative reciprocals of each other unless one is horizontal and the other is vertical. See [link] .

Section exercises

Verbal

What does it mean when we say that two lines are parallel?

It means they have the same slope.

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What is the relationship between the slopes of perpendicular lines (assuming neither is horizontal nor vertical)?

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How do we recognize when an equation, for example y = 4 x + 3 , will be a straight line (linear) when graphed?

The exponent of the x variable is 1. It is called a first-degree equation.

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What does it mean when we say that a linear equation is inconsistent?

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When solving the following equation:

2 x 5 = 4 x + 1

explain why we must exclude x = 5 and x = −1 as possible solutions from the solution set.

If we insert either value into the equation, they make an expression in the equation undefined (zero in the denominator).

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Algebraic

For the following exercises, solve the equation for x .

3 ( x + 2 ) 12 = 5 ( x + 1 )

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12 5 ( x + 3 ) = 2 x 5

x = 2 7

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x 3 3 4 = 2 x + 3 12

x = 6

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3 ( 2 x 1 ) + x = 5 x + 3

x = 3

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2 x 3 3 4 = x 6 + 21 4

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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