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Chapter 3: Problem 2

Clear the fractions, and rewrite each equation in standard form. \(y=5 x-9\)

Short Answer

Expert verified
5x - y = 9

Step by step solution

01

Identify the equation

The given equation is: y = 5x - 9This is in the form of a linear equation, where 'y' is isolated on one side.
02

Move all terms to one side

To rewrite the equation in standard form, which is Ax + By = C,move the term '5x' to the left side of the equation:y - 5x = -9
03

Arrange the equation

Standard form usually starts with the 'x' term followed by the 'y' term. So, rearrange the terms to:-5x + y = -9
04

Ensure standard form conventions

Standard form often prefers having 'A' as a positive integer. Multiply through by -1 to achieve this if needed:5x - y = 9

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Key Concepts

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

Standard Form
Standard form is a specific way of writing linear equations. The general structure is: Ax + By = C. Here, 'A', 'B', and 'C' are integers. This format is useful because it presents the equation in a clear and consistent manner. It helps in various applications such as graphing and solving systems of equations.

For an equation to be in standard form:
  • The 'x' and 'y' terms should be on the left side.
  • The constants should be on the right side.
  • 'A' should be a non-negative integer.

In the given exercise, the equation is converted into standard form from the form y = 5x - 9. Each step is designed to meet these conventions.
Clearing Fractions
Clearing fractions involves eliminating fractions from an equation to simplify calculations.

Here's how to do it:
  • Identify the fractions in your equation.
  • Find a common multiple for the denominators.
  • Multiply every term by this common multiple.

For example, if we had a fraction as part of our linear equation, such as y = (5/2)x - 3, we would multiply every term by 2 to clear the fraction:
2y = 5x - 6.
The result is an equation without fractions, making it easier to work with.
Although our example didn't initially have fractions, keeping this method in mind is useful for other similar problems.
Arranging Equations
Arranging equations properly ensures they meet the standard form and reduces error. Let's look at how to arrange equations:
  • Start by moving all variable terms to one side of the equation.
  • Keep the constant terms on the opposite side.
  • Ensure the standard form conventions are met, like having a positive 'A'.

Taking our exercise, starting from y = 5x - 9, we:
  • Move 5x to the left: y - 5x = -9.
  • Rearrange to get terms ordered properly: -5x + y = -9.
  • Ensure 'A' is positive by multiplying by -1: 5x - y = 9.

This final step makes sure the equation is neatly arranged in standard form.

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Most popular questions from this chapter

A high-speed Shinkansen train in Japan travels at a speed of \(\frac{270 \mathrm{~km}}{1 \mathrm{hr}}\) for \(18 \mathrm{~min}\). Find the distance it travels.

Use the slope formula to find the slope of the line that passes through the points. \((-6,-3) ;(-9,-2)\)

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(x-y=7\)

Use the slope formula to find the slope of the line that passes through the points. \(\left(\frac{5}{6}, \frac{1}{4}\right) ;\left(\frac{1}{6}, \frac{7}{4}\right)\)

Use the slope formula to find the slope of the line that passes through the points. \(\left(\frac{1}{3}, \frac{9}{7}\right) ;\left(\frac{5}{3}, \frac{3}{7}\right)\)
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