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Chapter 3: Problem 7

solve the equation. Show how to check your solution. $$\frac{3}{4}(x+6)=12$$

Short Answer

Expert verified
The solution to the equation is: x = 10.

Step by step solution

01

Simplify the equation

Multiply both sides by 4 to get rid of the fraction: \(4*\frac{3}{4}(x + 6) = 4*12\), which simplifies to \(3(x + 6) = 48\)
02

Isolate x

Now, expand the left side of the equation and subtract 18 from both sides: \(3x + 18 = 48\), which gives \(3x = 48 - 18 = 30\). Now, divide both sides by 3 to solve for x: \(x = 30/3 = 10\)
03

Check the solution

Substitute the value of x into the original equation: \(\frac{3}{4}(10+6) = 12\), which simplifies to \(\frac{3}{4}*16 = 12\). Simplifying the left side we get 12, which is equal to the right side. 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.

Fractions in Equations
Understanding how to deal with fractions in equations is crucial in algebra. When you see a fraction in an equation, it might look intimidating, but there are simple steps to handle it.

One effective method is to eliminate the fraction by multiplying every term by the denominator of the fraction. For example, in the equation \( \frac{3}{4}(x+6) = 12 \), we multiplied both sides by 4 to remove the fraction. This step transforms the equation into a simpler form, making it easier to solve.

Here are some steps to follow when solving equations with fractions:
  • Identify the fraction and its denominator.
  • Multiply all terms by the denominator to eliminate the fraction.
  • Simplify the equation and solve as usual.
Mastering this technique will help you handle similar algebraic equations with confidence.
Checking Solutions
After solving an equation, it's important to verify that your solution is correct. This process is known as checking the solution.

To check the solution, substitute your answer back into the original equation. In our example, after finding \(x = 10\), we substitute \(x\) with 10 in the equation \( \frac{3}{4}(x+6) = 12 \). You should perform the operations to ensure both sides of the equation are equal.

Here's a quick guide to checking your solution:
  • Substitute the solution back into the original equation.
  • Simplify both sides of the equation.
  • Confirm if the left-hand side equals the right-hand side.
If the sides match, you've most likely solved the equation correctly!
Algebraic Methods
Algebraic methods are essential tools for solving equations. They involve logical steps to manipulate equations and find unknown values.

In the problem we solved, we used the following algebraic methods:
  • Distributive property: We expanded \( 3(x + 6) \) to \( 3x + 18 \) by distributing the 3.
  • Isolating the variable: We subtracted 18 from both sides to isolate terms containing \(x\).
  • Solving for the variable: We divided by 3 to find \(x = 10\).
By understanding these algebraic techniques, you can efficiently approach and solve various types of algebraic equations. These skills are foundational and will support your mathematical growth.

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Most popular questions from this chapter

Use the following information. You open a snack stand at a fair. The income and expenses (in dollars) for selling each type of food are shown in the matrices. \(\begin{array}{ccccccc}\text { Day 1 } & \text { Income } & \text { Expenses } & \text { Day 2 } & \text { Income } & \text { Expenses } \\ \text { Hamburgers } & 72 & 14 & \text { Hamburgers } & 62 & 10 \\ \text { Hot dogs } & 85 & 18 & \text { Hot dogs } & 52 & 11 \\ \text { Tacos } & 46 & 19 & \text { Tacos } & 72 & 26\end{array}\) What were your total income and expenses for selling each type of food for the two days of the fair?

Find the percent. Round to the nearest whole percent. \(\$ 2.25\) tip on a cab fare of \(\$ 14\)

Find the resulting unit of measure. (miles) \(\div\) (miles per hour)

There are 4 times as many nickels as dimes in a coin bank. The coins have a total value of 600 cents (6.00 dollar) Find the number of nickels.

Round the result to the nearest hundredth. Check the rounded solution. $$2(4-3 x)=-x$$
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