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Chapter 6: Problem 1

Solve each equation for \(y\). $$\frac{y-5}{x+3}=-\frac{4}{3}$$

Short Answer

Expert verified
y = -\frac{4}{3}x + 1

Step by step solution

01

- Understand the Equation

We are given the equation \(\frac{y-5}{x+3} = -\frac{4}{3}\). We aim to solve for the variable \(y\).
02

- Clear the Fraction

Multiply both sides of the equation by \(x + 3\) to clear the fraction. This gives us \((y - 5) = -\frac{4}{3} \times (x + 3)\).
03

- Distribute the Constant

Distribute \(-\frac{4}{3}\) on the right-hand side of the equation: \(y - 5 = -\frac{4}{3}x - 4\).
04

- Isolate the Variable \(y\)

Add 5 to both sides to isolate \(y\): \(y = -\frac{4}{3}x - 4 + 5\).
05

- Simplify

Combine the constant terms on the right-hand side: \(y = -\frac{4}{3}x + 1\).

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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 is an algebraic expression where the highest power of the variable is one. This means that the variable is not squared or raised to any higher power. Linear equations often look like this: \ ax + by = c \ where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables.
  • They graph to a straight line.
  • They have constant rates of change.
It's important to recognize linear equations because they are easier to solve than non-linear equations.
fractions
Fractions represent parts of a whole. They are written as \(\frac{numerator}{denominator}\). When working with fractions in equations:
  • Clear the fraction by multiplying every term by the denominator.
  • Simplify the resulting equation.
In our exercise, we have the fraction: \ \frac{y-5}{x+3} = -\frac{4}{3} \ To solve it:
  1. We multiply both sides by \(x + 3\) to eliminate the denominator.
  2. This leaves us with a simpler equation to solve.
variables manipulation
Manipulating variables is about rearranging equations to isolate the variable you are solving for. This involves:
  • Addition and subtraction to move terms from one side to the other.
  • Multiplication and division to simplify or clear coefficients.
For the exercise given:
  1. We distribute \(-\frac{4}{3}\) across \(x + 3\).
  2. Then, add or subtract terms to get \(y\) alone.
This manipulation helps reveal the value of the variable we are interested in.
isolation of variables
Isolating variables means getting a single variable by itself on one side of the equation. This is critical in solving for that variable:
\ \frac{y - 5}{x + 3} = -\frac{4}{3} \
  • First, we clear the fraction by multiplying both sides by \(x + 3\).
  • Next, distribute \(-\frac{4}{3}\): \(y - 5 = -\frac{4}{3}x - 4\).
  • Finally, add 5 to both sides: \(y = -\frac{4}{3}x + 1\).
Now, \(y\) is isolated, and the problem is solved.

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Most popular questions from this chapter

Solve each equation. $$\frac{2 m}{5}=\frac{10}{m}$$

Distance traveled. Bonita drove 100 miles in \(x\) hours. Assuming she continued to drive at the same speed, write a rational expression for the distance that she traveled in the next \(\frac{3}{4}\) of an hour.

Solve each equation. Identify each equation as a conditional equation, an inconsistent equation, or an identity. State the solution sets to the identities using interval notation. $$\frac{2}{x+1}+\frac{3}{x-1}=\frac{5 x+2}{x^{2}-1}$$

Perform the indicated operations. When possible write down only the answer. $$\frac{x+3}{\frac{1}{3}}$$

Perform the indicated operations. When possible write down only the answer. $$\left(x^{2}+y^{2}\right) \div(-1)$$
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