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Chapter 2: Problem 42

Find \(y\) given that \(x=3\). $$-3 x-4 y=4$$

Short Answer

Expert verified
y = -\frac{13}{4}

Step by step solution

01

Substitute the value of x

Given the equation -3x - 4y = 4 and that x = 3, substitute 3 for x: $$-3(3) - 4y = 4$$
02

Simplify the equation

Calculate -3(3) to get -9: $$-9 - 4y = 4$$
03

Isolate the variable term

Add 9 to both sides of the equation to isolate the term containing y: $$-4y = 4 + 9$$ Calculate the sum on the right side: $$-4y = 13$$
04

Solve for y

Divide both sides by -4 to solve for y: $$y = \frac{13}{-4}$$ Simplify the fraction: $$y = -\frac{13}{4}$$

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Key Concepts

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

Substitution Method
The substitution method is a key technique in solving linear equations. This method involves replacing a variable with a given value to simplify the equation.

In our example, we are given the equation \(-3x - 4y = 4\) and the value \(x = 3\). To apply the substitution method:
  • Substitute 3 for \x\ in the equation: \(-3(3) - 4y = 4\).
  • The equation now only contains the variable \y\, making it easier to solve.
This method is very effective when one variable is already known.
Isolating Variables
Isolating variables is a crucial step in solving equations. The goal is to manipulate the equation such that we have the variable of interest on one side of the equation by itself.

In our example, after substituting 3 for \x\, we have: \(-9 - 4y = 4\).

Next:
  • Add 9 to both sides to isolate the term with \y\: \(-9 + 9 - 4y = 4 + 9\). This simplifies to \(-4y = 13\).
Isolating the variable helps simplify the equation into a solvable form.
Simplifying Fractions
Simplifying fractions is an essential skill in algebra to present the final answer in its simplest form.

Let's continue from where we isolated the variable: \-4y = 13\.
  • To solve for \y\, divide both sides by \(-4)\: \y = \frac{13}{-4}\.
  • We can simplify this fraction by realizing that dividing by a negative number results in a negative fraction: \y = -\frac{13}{4}\).
Simplifying fractions ensures the solution is clear and easy to understand.

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Most popular questions from this chapter

Write each union or intersection of intervals as a single interval if possible. $$[1,4) \cup(2,6]$$

Solve each inequality. Write the solution set using interval notation. $$3-5|x|>-2$$

Discussion. In each case, write the resulting set of numbers in interval notation. Explain your answers. a) Every number in \((3,8)\) is multiplied by 4 b) Every number in \([-2,4)\) is multiplied by \(-5\) c) Three is added to every number in \((-3,6)\) d) Every number in \([3,9]\) is divided by \(-3 .\)

Show a complete solution to each problem. Really odd integers. If the smaller of two consecutive odd integers is subtracted from twice the larger one, then the result is \(13 .\) Find the integers.

Solve problem by using an inequality. Sewing machines. Charles wants to buy a sewing machine in a city with a \(10 \%\) sales tax. He has at most \(\$ 700\) to spend. In what price range should he look?
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