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Chapter 8: Problem 56

Find the equation of each line. Write the equation using standard notation unless indicated otherwise. $$ \text { Slope }-\frac{3}{5} \text { ; through }(4,-1) $$

Short Answer

Expert verified
The equation of the line is \(-3x + 5y = 7\).

Step by step solution

01

Identify the Slope-Intercept Format

The slope-intercept form of the equation of a line is given by \( y = mx + c \), where \( m \) is the slope of the line and \( c \) is the y-intercept.
02

Substitute the Slope and Point into the Equation

Given that the slope \( m = -\frac{3}{5} \) and the point is \((4, -1)\), substitute these into the slope-intercept form: \(-1 = -\frac{3}{5}(4) + c\).
03

Solve for the Y-Intercept

Calculate \(-\frac{3}{5} \times 4 = -\frac{12}{5}\). Substitute this into the equation to find \( c \): \[-1 = -\frac{12}{5} + c\].Convert \(-1\) into a fraction with the same denominator: \(-\frac{5}{5}\). Solve for \( c \): \[c = -1 + \frac{12}{5} = \frac{7}{5}\].
04

Write the Equation in Slope-Intercept Form

Now that we know \( c = \frac{7}{5} \), plug \( m \) and \( c \) back into the equation: \[y = -\frac{3}{5}x + \frac{7}{5}\].
05

Convert to Standard Form

The standard form of the line's equation is \( Ax + By = C \). Rearrange the slope-intercept equation into standard form: Start with \( -\frac{3}{5}x + y = \frac{7}{5}\). Multiply the entire equation by 5 to clear fractions: \(-3x + 5y = 7\).

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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 of a linear equation is a very popular way to express the equation of a line. It is written as \( y = mx + c \). Here, \( m \) stands for the slope of the line, which tells us how steeply the line inclines or declines. The \( c \) represents the y-intercept, which is the point where the line crosses the y-axis.
When working with the slope-intercept form:
  • The value of \( m \) determines the direction and steepness of the line. A positive \( m \) means an upward slope, while a negative \( m \) indicates a downward slope.
  • The value of \( c \) gives a clear indication of where the line intersects the y-axis, which means when \( x = 0 \).
Using this form simplifies the process of graphing lines and provides a direct relationship between the slope and intercept values.
Standard Form
The standard form of a linear equation is expressed as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) and \( B \) should not both be zero. This form is very structured, which makes it useful in situations where you need to find intercepts or manipulate equation parts.
A key advantage of the standard form is:
  • It's easy to solve for both x- and y-intercepts by plugging zero in for the other variable.
  • All terms are compared directly on one side, which assists in solving equations simultaneously or analyzing relationships between lines.
Typical steps to convert from slope-intercept form to standard form involve rearranging the equation: start by eliminating fractions if any, and ending with all variables on one side and the constant on the other.
Solving for Y-Intercept
Finding the y-intercept \( c \) in the equation \( y = mx + c \) involves substituting a known slope and a point the line passes through. If you have a slope \( m \) and a point \((x_1, y_1)\), plug them into the slope-intercept form: \(y_1 = mx_1 + c\). Then solve for \( c \).
For example, given a slope \( m = -\frac{3}{5} \) and the point \((4, -1)\):
  • Substitute into the equation: \(-1 = -\frac{3}{5}(4) + c\).
  • Calculate the multiplication: \(-\frac{3}{5} \times 4 = -\frac{12}{5}\).
  • Solve for \( c \) by isolating it: \(c = -1 + \frac{12}{5} = \frac{7}{5}\).
This process identifies the point where the line intersects the y-axis.
Rearranging Equations
Rearranging involves transforming one form of an equation to another while maintaining its equality. It's crucial when you need to express an equation in a specific format, like turning a slope-intercept form into a standard form.
Here's a basic approach to rearranging equations:
  • Identify your starting equation and your target format. For example, starting with \( y = mx + c \) and changing it to \( Ax + By = C \).
  • Move all terms involving \( x \) and \( y \) to one side by performing operations like addition, subtraction, multiplication, or division that transform the equation correctly.
  • Simplify the equation by clearing out fractions or decimals. Here, multiplying through by a common denominator helps eliminate fractions, leading to integer coefficients.
  • Check your work by ensuring that all terms are balanced and the equation remains true to its original relationships.
By mastering these skills, one can maneuver between different forms effortlessly, an essential skill in algebra.

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