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Chapter 7: Problem 6

Write the equation \(10 x-7 y=70\) in slope-intercept form.

Short Answer

Expert verified
The 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 line's equation is given by \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.
02

Rearrange the original equation

Start with the original equation: \(10x - 7y = 70\). The goal is to solve for \(y\) in terms of \(x\).
03

Isolate the term involving \(y\)

Subtract \(10x\) from both sides of the equation: \(-7y = -10x + 70\).
04

Solve for \(y\)

Divide each term by \(-7\) to solve for \(y\): \(y = \frac{10}{7}x - 10\).
05

Write the final equation

Now, the equation is in slope-intercept form: \(y = \frac{10}{7}x - 10\).

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Key Concepts

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

linear equations
Linear equations are mathematical expressions that describe a straight line when plotted on a graph. The general form of a linear equation is either in the form of \[Ax + By = C\] or \[y = mx + b\].

Here:
  • \(A\) and \(B\) are coefficients.
  • \(x\) and \(y\) are variables representing horizontal and vertical coordinates respectively.
  • \(C\) is the constant term.

The slope-intercept form \(y = mx + b\) is a convenient format as it easily shows the slope \(m\), which indicates the line's steepness, and the y-intercept \(b\), where the line touches the y-axis.
The linear equation \(10x - 7y = 70\) is not initially in this format, but it can be transformed to reveal useful information about the line's slope and y-intercept.
solving for y
Solving for \(y\) means rearranging the equation so that \(y\) is isolated on one side of the equation. To achieve the slope-intercept form from \(10x - 7y = 70\), follow these steps:
  • Start with the given equation: \(10x - 7y = 70\).
  • Subtract \(10x\) from both sides to isolate the \(y\) term: \(-7y = -10x + 70\).
  • Divide every term by \(-7\) to solve for \(y\): \(y = \frac{10}{7}x - 10\).

Now, \(y\) is on one side and the equation is expressed in terms of \(x\), illuminating the slope and y-intercept directly.
algebraic manipulation
Algebraic manipulation involves using algebra rules to rearrange and simplify an equation. In our equation \(10x - 7y = 70\), key manipulations include:
  • Transposing terms:Moving \(10x\) to the other side by subtraction.

  • Isolating the variable \(y\):Getting \(y\) alone by dividing by the coefficient associated with \(y\)
Understanding these steps is essential as they form the basis for solving more complicated equations and algebraic expressions. The end result, \(y = \frac{10}{7}x - 10\), is cleaner and more functional for graphing and interpreting the linear relationship between \(x\) and \(y\).

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