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Chapter 6: Problem 3

Solve each equation and check the solution. $$\frac{z-3}{2}=z+2$$

Short Answer

Expert verified
The solution to the equation \(\frac{z-3}{2}=z+2\) is \( z = -7\).

Step by step solution

01

Remove Fraction

The equation has a fraction on one side, so it needs to be eliminated first. To do this, multiply both sides by the denominator of the fraction. Here, the denominator is 2, so the equation becomes: \(2 \cdot \frac{z-3}{2} = 2 \cdot (z+2)\), which simplifies to \(z - 3 = 2z + 4\).
02

Rearrange and Solve Equation

Subtract \(z\) from both sides to get the equation \( -3 = z + 4\). After then, subtract 4 from both sides to get: \( -3 - 4 = z\), which simplifies to \( z = -7\).
03

Check the Solution

Substitute the value of \(z\) obtained in the original given equation as follows: \(\frac{-7 - 3}{2} = -7 + 2\). Simplify the equation to check if both sides are equal. The left-hand side simplifies to \(-5\) and the right-hand side simplifies to \(-5\) also. Both sides are equal, therefore the solution is correct.

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Key Concepts

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

Solving Linear Equations
Linear equations are fundamental in algebra. They allow us to find the value of an unknown variable. To solve a linear equation, you need to rearrange the equation so that the variable is on one side and all other terms are on the other. This involves performing operations such as addition, subtraction, multiplication, or division on both sides of the equation.

Let's break it down further:
  • Look at the equation and identify where the variable is located.
  • Use operations to isolate the variable on one side of the equation. This might involve moving terms by adding or subtracting them from both sides.
  • Sometimes it's necessary to multiply or divide each term by a number to simplify the equation further.
The aim is to have the variable equal to a number, indicating its value. Once you isolate the variable successfully, the equation is solved.
Checking Solutions in Algebra
After solving a linear equation, it鈥檚 crucial to check your solution. This guarantees that your computed value for the variable satisfies the original equation. If the two sides of the equation are equal when the variable is substituted back in, then the solution is correct.

Here鈥檚 how you can check the solution:
  • Substitute the solved value of the variable back into the original equation.
  • Simplify both sides of the equation separately.
  • Ensure both sides of the equation equal the same numerical value.
If everything matches up, you have verified your solution!
Removing Fractions from Equations
Fractions in an equation make it look complex, but removing them simplifies the task of solving for the variable. This involves multiplying every term of the equation by the denominator of the fraction, thereby eliminating the fraction.

Let's see this process:
  • Identify the fraction in the equation and note its denominator.
  • Multiply both sides of the equation by this common denominator. This will cancel out the fraction.
  • After the fraction is gone, you'll often have a simpler linear equation to solve.
This step is key in making the equation easier to handle, paving the way for straightforward solutions.

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Most popular questions from this chapter

Let \(x\) equal a number of your choosing. Without simplifying first, use a calculator to evaluate $$\frac{x^{2}+x-6}{x^{2}+3 x} \cdot \frac{x^{2}}{x-2}$$ Try again, with a different value of \(x\). If you were to simplify the expression, what do you think you would get?

Explain how you would decide what to do first when you solve an equation that involves fractions.

$$\text { Solve: } x^{-2}+x^{-1}=0$$.

Set up and solve a proportion. A cologne can be made by mixing 3 drops of pure essence with 7 drops of distilled water. How many drops of water should be used with 42 drops of pure essence?

Set up and solve a proportion. A recipe for chocolate chip cookies calls for \(1 \frac{1}{4}\) cups of flour and 1 cup of sugar. The recipe will make \(3 \frac{1}{2}\) dozen cookies. How many cups of flour will be needed to make 12 dozen cookies?
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