Question

In Problems 45-48, find the coordinates of the point of intersection. Then write an equation for the line through that point perpendicular to the line given first. \(4 x-5 y=8\) \(2 x+y=-10\)

Short answer

The point of intersection is (-3, -4) and the perpendicular line equation is \(y = -\frac{5}{4}x - \frac{31}{4}\).

Step-by-step solution

Solve for y in both equations

To find the point of intersection, we need both equations in the form of y.First equation: \(4x - 5y = 8\) becomes \(y = \frac{4}{5}x - \frac{8}{5}\).Second equation: \(2x + y = -10\) becomes \(y = -2x - 10\).

Set equations equal and solve for x

Now set \(\frac{4}{5}x - \frac{8}{5}\) equal to \(-2x - 10\) and solve for x.\[ \frac{4}{5}x - \frac{8}{5} = -2x - 10 \] Multiply entire equation by 5 to eliminate fractions:\[ 4x - 8 = -10x - 50 \] Add \(10x\) to both sides:\[ 14x - 8 = -50 \] Add 8 to both sides:\[ 14x = -42 \] Divide by 14:\[ x = -3 \]

Substitute x back into either equation to find y

Substitute \(x = -3\) back into either equation to find the y-coordinate. Using \(y = -2x - 10\):\[ y = -2(-3) - 10 \] \[ y = 6 - 10 \] \[ y = -4 \] So, the point of intersection is \((-3, -4)\).

Find the slope of the first line

We need the slope of the first line equation to find the perpendicular line's slope. For \(4x - 5y = 8\), the slope is \(\frac{4}{5}\) from \(y = \frac{4}{5}x - \frac{8}{5}\).

Determine the slope of the perpendicular line

The slope of a line perpendicular to another is the negative reciprocal of the original line's slope.The negative reciprocal of \(\frac{4}{5}\) is \(-\frac{5}{4}\).

Write the equation of the perpendicular line

Use the point of intersection \((-3, -4)\) and the perpendicular slope \(-\frac{5}{4}\) to form the line equation \(y = mx + c\).\[ -4 = -\frac{5}{4}(-3) + c \] \[ -4 = \frac{15}{4} + c \] Subtract \(\frac{15}{4}\) from \(-4\):\[ c = -\frac{31}{4} \] Thus, the equation is \(y = -\frac{5}{4}x - \frac{31}{4}\).

Key concepts

Perpendicular Lines

Perpendicular lines are two lines that intersect at a right angle, meaning the angle between them is 90 degrees. To determine if two lines are perpendicular, you need to look at their slopes. In coordinate geometry, the slopes of perpendicular lines have a special relationship. They are negative reciprocals of each other. For example, if one line has a slope of \( \frac{4}{5} \), then a line perpendicular to it will have a slope of \( -\frac{5}{4} \). You can find the negative reciprocal of a slope by flipping the fraction and changing its sign.
When you are given a line and asked to find a line perpendicular to it that passes through a certain point, you can use the point-slope form of the line equation. This would be \( y = mx + c \), where \( m \) is the slope, and \( c \) is the y-intercept. Once you calculate the slope, you can substitute the coordinates of the given point to find the equation of your perpendicular line.

System of Equations

A system of equations consists of multiple equations that are solved together. The goal is to find values of variables that satisfy all equations in the system simultaneously. In many problems, such as finding the intersection of two lines, you deal with a system of two linear equations. The lines intersect where both equations are true at the same time.
To solve for the intersection, you can use the substitution method or the elimination method. In the original problem, both equations are first solved for \( y \) to make them easier to work with. Next, the substitution method is used, where one equation is substituted into the other. This lets you solve for one variable at a time. Once you find one variable's value, it can be plugged back into any of the original equations to find the value of the other variable.

• The substitution method is useful when one equation can be easily solved for one variable.
• The elimination method is often quicker if you can easily add or subtract equations to eliminate a variable.

Coordinate Geometry

Coordinate geometry, or analytic geometry, links algebra and geometry through the use of coordinates on graphs. It allows for a visual representation of algebraic equations and relationships, making it easier to understand intersections, slopes, and distances.
A primary concept in coordinate geometry is the coordinate plane, usually composed of an x-axis (horizontal) and a y-axis (vertical). Every point on this plane can be represented by an ordered pair \((x, y)\). The equation of a line, such as \(4x - 5y = 8\), describes every point that lies on that line. Equations are often converted into slope-intercept form \( y = mx + c \) to better understand slope and intercepts.
Understanding the layout of the coordinate plane helps solve geometric problems involving intersections and distances.

• Each equation represents a line on the plane.
• Intersection points are where lines cross each other.

Line Equations

Line equations are mathematical expressions that describe the relationship between x and y coordinates on a line. They enable us to represent lines algebraically and geometrically. The most common form of a line equation is the slope-intercept form: \( y = mx + c \), where \( m \) is the slope of the line, and \( c \) is the y-intercept.
The slope is a measure of the steepness of the line, calculated as the 'rise over run,' or the change in y divided by the change in x. The y-intercept is the point where the line crosses the y-axis.
Other forms of line equations include the point-slope form \( y - y_1 = m(x - x_1) \) and the standard form \( Ax + By = C \), each useful in different scenarios. Understanding how to manipulate these forms and calculate their components enables problem-solving in both pure and applied mathematics.

• Equations can be transformed into different forms by algebraic manipulation.
• Slope-intercept form is particularly handy for quickly identifying a line’s slope and y-intercept.