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Chapter 3: Problem 4

Solve each equation. $$7 x+3=24$$

Short Answer

Expert verified
The solution is \(x = 3\).

Step by step solution

01

Isolate the Term with the Variable

First, we need to isolate the term with the variable on one side of the equation. Start by subtracting 3 from both sides of the equation: \[ 7x + 3 - 3 = 24 - 3 \] which simplifies to \[ 7x = 21 \].
02

Solve for the Variable

Now that we have \(7x = 21\), we need to solve for \(x\). This is done by dividing both sides of the equation by 7: \[ \frac{7x}{7} = \frac{21}{7} \] resulting in \(x = 3\).

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Key Concepts

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

Solving Linear Equations
In mathematics, solving linear equations is a fundamental process. A linear equation is an equation between two variables that produces a straight line when plotted on a graph. Our goal when solving such equations is to find the value of the unknown variable that makes the equation true. In our example, we have the equation \(7x + 3 = 24\), where our task is to find the value of \(x\). Solving equations typically involves a few basic steps:
  • Identify the equation you need to solve.
  • Manipulate the equation to isolate the variable you're solving for.
  • Perform the necessary operations to determine the value of the variable.
It's important to perform the same operation on both sides of the equation to maintain equality. Each step should bring you closer to the solution.
Isolating the Variable
When solving linear equations, 'isolate the variable' is a key strategy. This simply means rearranging the equation so that the variable is on one side of the equation by itself. In our example, we start with \(7x + 3 = 24\). The first step to isolate \(x\) is to eliminate any constants from the side of the equation that \(x\) is on. Here鈥檚 how to do it:
  • Subtract 3 from both sides of the equation: \(7x + 3 - 3 = 24 - 3\).
  • This simplifies to \(7x = 21\).
By subtracting, we have effectively moved all other terms to the opposite side of the equation, leaving us with the variable \(x\) alone with its coefficient.
Equation Simplification
Equation simplification involves performing operations to simplify the equation, making it easier to solve. After isolating the variable, we simplify the equation by solving the remaining terms. In this case, we are left with \(7x = 21\). To solve for \(x\), we need to simplify further by removing the coefficient of \(x\).
  • Divide both sides by 7: \(\frac{7x}{7} = \frac{21}{7}\).
  • This simplification leads us to the solution \(x = 3\).
By simplifying, we reduce the complexity of the equation. This makes it easier to see the solution clearly and ensures that no arithmetic errors are made. Always double-check your work to confirm that your solution is correct.

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Most popular questions from this chapter

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