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Chapter 2: Problem 6

Provide the appropriate response. 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 Slope-Intercept Form

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

- Isolate the y-term

Start with the given equation: \[ 10x - 7y = 70 \] The goal is to isolate \(y\) on one side. First, subtract \(10x\) from both sides of the equation: \[ -7y = -10x + 70 \]
03

- Solve for y

Next, divide every term by \(-7\) to solve for \(y\): \[ y = \frac{-10x}{-7} + \frac{70}{-7} \] Simplify the fractions: \[ y = \frac{10}{7}x - 10 \] Now the equation is in slope-intercept form.

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Key Concepts

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

Linear Equations
Linear equations involve two variables and they graph as straight lines. They appear in the form \(ax + by = c\) where \(a\), \(b\), and \(c\) are constants. In our exercise, the given linear equation is \[10x - 7y = 70\]. To better understand this, we'll convert it to the slope-intercept form.
Isolation Method
The isolation method focuses on getting one variable alone on one side of an equation. To convert our given equation \[10x - 7y = 70\] into slope-intercept form, we need to isolate \(y\).

Follow these steps:
  • First, isolate the \(y\)-term by moving other terms to the opposite side. From our equation, subtract \(10x\) from both sides to get: \[ -7y = -10x + 70 \]
  • Next, solve for \(y\) by dividing every term by \(-7\). This step is crucial as it'll simplify the equation. We get: \[ y = \frac{-10x}{-7} + \frac{70}{-7} \]
This process ensures \(y\) is isolated and the equation is gradually transformed toward the slope-intercept form.
Simplifying Fractions
In many algebra problems, especially when converting equations, simplifying fractions is vital. After isolating the variable in our example, we have the expression \[ y = \frac{-10x}{-7} + \frac{70}{-7} \].

To simplify:
  • First, simplify \(\frac{-10x}{-7}\). The negatives cancel each other out, resulting in: \[ \frac{10}{7}x \]
  • Next, simplify \(\frac{70}{-7}\), which equals \-10\.
This gives us the final equation in slope-intercept form: \[ y = \frac{10}{7}x - 10 \].
Simplifying fractions helps to make equations more understandable and easier to interpret.

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Most popular questions from this chapter

An equation that defines \(y\) as a function \(f\) of \(x\) is given. (a) Solve for \(y\) in terms of \(x\), and write each equation using function notation \(f(x) .\) (b) Find \(f(3)\). $$ y+2 x^{2}=3 $$

Rental on a car is \(\$ 150,\) plus \(\$ 0.50\) per mile. Let \(x\) represent the number of miles driven and \(f(x)\) represent the total cost to rent the car. (a) Write a linear function that models this situation. (b) Find \(f(250)\). Interpret the answer in the context of this problem. (c) Find the value of \(x\) if \(f(x)=400\). Express this situation using function notation, and interpret it in the context of this problem.

Find the slope of each line, and sketch its graph. \(x+2=0\)

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Graph the union of each pair of inequalities. $$ x-2>y \text { or } x<1 $$
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