/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 250 Solve each linear equation. $$... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 250

Solve each linear equation. $$\frac{2}{3}(9 c-3)=22$$

Short Answer

Expert verified
c = 4

Step by step solution

01

Distribute the fraction

Distribute \(\frac{2}{3}\) to both terms inside the parentheses: \[ \frac{2}{3} \times (9c - 3) = \frac{2}{3} \times 9c - \frac{2}{3} \times 3 \] Simplify the fractions: \[ 6c - 2 = 22 \]
02

Isolate the variable term

Add 2 to both sides to isolate the term containing the variable: \[ 6c - 2 + 2 = 22 + 2 \] Simplify: \[ 6c = 24 \]
03

Solve for the variable

Divide both sides by 6 to solve for \c: \[ c = \frac{24}{6} \] Simplify: \[ c = 4 \]

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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 mathematical sentences that show the equality of two expressions separated by an equals sign. The goal is to find the value of the variable that makes the equation true.
For example, consider the equation \(\frac{2}{3}(9 c-3)=22\). To solve this, you must perform several systematic steps. These steps typically involve simplifying both sides of the equation, using various mathematical properties, isolating the variable, and then solving for it. Linear equations can have different forms, but the same principles and methods are applied to solve them.
distributive property
The distributive property is a fundamental concept in algebra used to simplify expressions. It states that \(a(b + c) = ab + ac \); you 'distribute' the multiplication over addition or subtraction within parentheses.
In our example, we start with the equation \(\frac{2}{3}(9 c-3)=22\). Using the distributive property, we multiply \(\frac{2}{3}\) by each term inside the parentheses: \(\frac{2}{3} \times 9c - \frac{2}{3} \times 3\). This results in \(6c - 2\). Applying the distributive property early on helps to simplify and make the equation more manageable.
isolating variables
Isolating the variable means manipulating the equation to get the variable on one side and the constants on the other. This helps to solve for the variable.
Continuing from our example equation \(6c - 2 = 22\), we need to isolate the term containing the variable (\(c\)). We do this by adding 2 to both sides of the equation, resulting in \(6c - 2 + 2 = 22 + 2\), which simplifies to \(6c = 24\). Now the variable term is isolated, and we can proceed to solve for \(c\).
simplifying fractions
Simplifying fractions involves reducing them to their simplest form. This can make solving linear equations cleaner and easier.
For instance, in the example \(6c = 24\), to find \(c\), you divide both sides by 6: \(c = \frac{24}{6}\), which simplifies to \(c = 4\). Simplifying fractions is often the final step in solving the equation, ensuring the solution is as straightforward as possible.
Remember, whenever you encounter fractions in equations, break them down into simpler terms to help make the arithmetic manageable and the solution clearer.

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