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Chapter 7: Problem 6

In Exercises \(1-6,\) provide the appropriate response. Write the equation \(10 x-7 y=70\) in slope-intercept form.

Short Answer

Expert verified
The equation in slope-intercept form is \( y = \frac{10}{7}x - 10 \).

Step by step solution

01

Understand the Slope-Intercept Form

The slope-intercept form of a linear equation is: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept.
02

Isolate the y-Term

Begin with the given equation: \[ 10x - 7y = 70 \]Subtract \( 10x \) from both sides:\[ -7y = -10x + 70 \]
03

Solve for y

Divide every term by \( -7 \):\[ y = \frac{-10x + 70}{-7} \]Simplify the equation:\[ y = \frac{10}{7}x - 10 \]
04

Write the Final Equation

The equation in slope-intercept form is:\[ y = \frac{10}{7}x - 10 \]

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Key Concepts

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

Linear Equations
A linear equation describes a straight line on a graph. It鈥檚 an equation in which the highest power of the variable is one. Linear equations are usually written in the form of Ax + By = C. For example, the equation 10x - 7y = 70 is a linear equation. Linear equations can be written in different forms, such as the standard form and the slope-intercept form:
  • Standard form: Ax + By = C
  • Slope-intercept form: y = mx + b
Understanding how to transform between these forms is crucial in algebra.
Solving for y
Solving for y means getting y by itself on one side of the equation. This is an important step when converting a linear equation to the slope-intercept form. Begin with the equation you are given and focus on isolating the y-term. For the equation 10x - 7y = 70, follow these steps:
  • Subtract 10x from both sides: -7y = -10x + 70
  • Divide every term by -7: \[ y = \frac{-10x + 70}{-7} \]
Simplify the equation to find the value of y.
Isolating Variables
Isolating variables is the process of getting a specific variable on its own on one side of the equation. This often involves several algebraic steps like addition, subtraction, multiplication, or division. In our example, to isolate y in 10x - 7y = 70, you need to:
  • Subtract 10x from both sides to remove the x term: -7y = -10x + 70
  • Divide all terms by -7 to get y alone: \[ y = \frac{-10x + 70}{-7} = \frac{10}{7}x - 10 \]
Isolating y helps you convert the equation to slope-intercept form.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations to solve for a variable. This requires a good understanding of the principles of addition, subtraction, multiplication, and division. To transform 10x - 7y = 70:
  • Start by subtracting 10x from both sides, to move the x term away from the y term. This gives us: -7y = -10x + 70
  • Next, divide all terms by -7 to simplify: \[ y = \frac{-10x + 70}{-7} = \frac{10}{7} x - 10 \]
This step-by-step manipulation helps us get from the standard form to the slope-intercept form.

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