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Chapter 1: Problem 61

Exer. \(61-66:\) Find an equation of the line that satisfies the given conditions. Through \(A(5,-3) ;\) slope -4

Short Answer

Expert verified
The equation is \( y = -4x + 17 \).

Step by step solution

01

Understand the Slope-Intercept Form

The slope-intercept form of a line's equation is given by the formula \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. In this problem, the slope \( m \) is given as \(-4\).
02

Apply Point-Slope Formula

To find the equation of the line through a given point with a specified slope, we can use the point-slope formula: \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1) = (5, -3)\) and \(m = -4\).
03

Substitute Values into the Point-Slope Formula

Substitute the values into the point-slope formula: \( y - (-3) = -4(x - 5) \), which simplifies to \( y + 3 = -4(x - 5) \).
04

Simplify the Equation

Distribute the slope \(-4\) on the right side: \( y + 3 = -4x + 20 \).
05

Solve for y in Slope-Intercept Form

Finally, solve for \( y \) to put the equation in slope-intercept form: \( y = -4x + 20 - 3 \), which simplifies to \( y = -4x + 17 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form is a straightforward way to express the equation of a line. It is written as \( y = mx + c \). Here, \( m \) represents the slope of the line, which tells us how steep the line is. The \( c \) is the y-intercept, which is the point where the line crosses the y-axis.
  • If the slope \( m \) is positive, the line rises as it moves from left to right.
  • If the slope is negative, like \(-4\) in this exercise, the line falls as it moves from left to right.
  • The y-intercept \( c \) is the value of \( y \) when \( x = 0 \).
For the exercise, we found that the line's equation in the slope-intercept form is \( y = -4x + 17 \). This equation tells us that for every one unit increase in \( x \), \( y \) decreases by 4 units.
Point-Slope Formula
The point-slope formula is another useful method for writing the equation of a line. It's especially helpful when you know a point on the line and the slope. The formula is written as \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a specific point on the line and \( m \) is the slope.
  • This formula emphasizes the change from a known point, \( y_1 \), based on the distance \( x \) travels from \( x_1 \).
  • It effectively connects the known point and slope to formulate the line's behavior.
In the given exercise, we used the point-slope formula with point \( (5, -3) \) and slope \( -4 \). We substituted these values into the formula to obtain the equation \( y + 3 = -4(x - 5) \). This equation was then further simplified.
Linear Equation
A linear equation represents a straight line on a graph. Its general form is often written as \( ax + by = c \), which can be rewritten in slope-intercept form or solved to find different attributes of the line. Linear equations are simple yet powerful tools in mathematics.
  • The equation represents a set of points that make up a line in the Cartesian plane.
  • They play a crucial role in coordinate geometry, helping to explain relationships between coefficients and graph characteristics.
  • Every linear equation shows up as a straight line.
In our exercise, the final linear equation we derived was \( y = -4x + 17 \), perfectly illustrating how these assumptions and transformations work together to describe a line's path through coordinate space.

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