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Chapter 8: Problem 2

Solve equation. Check your solution. \(4 x+9=7 x\)

Short Answer

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

Step by step solution

01

Rearrange the Equation

To solve the equation \(4x + 9 = 7x\), the first step is to bring all terms involving the variable \(x\) to one side of the equation. Subtract \(4x\) from both sides to obtain:\[4x + 9 - 4x = 7x - 4x\]This simplifies to:\[9 = 3x\]
02

Solve for x

Next, we want to isolate \(x\). Divide both sides of the equation \(9 = 3x\) by 3 to solve for \(x\):\[x = \frac{9}{3}\]This simplifies to:\[x = 3\]
03

Check the Solution

To verify the solution, substitute \(x = 3\) back into the original equation \(4x + 9 = 7x\):Substituting \(x = 3\), the left-hand side becomes:\[4(3) + 9 = 12 + 9 = 21\]The right-hand side becomes:\[7(3) = 21\]Since both sides are equal, the solution \(x = 3\) satisfies the original equation.

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Key Concepts

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

Checking Solutions
When solving linear equations, it's crucial to verify if the solution actually satisfies the original equation. This process ensures that no mistakes were made and that the solution is valid. After finding a solution, substitute the value back into the original equation and perform the calculations to check if both sides of the equation balance.
For example, in the equation \(4x + 9 = 7x\), after determining that \(x = 3\), you substitute \(x = 3\) back into the equation. The left-hand side becomes \(4(3) + 9\), which calculates to \(21\). The right-hand side, \(7(3)\), also equals \(21\).
  • If both sides of the equation are equal, then the solution is correct.
  • If they are not equal, you need to go back and check your calculations.
Checking solutions is a great way to build confidence in your math skills!
Rearranging Equations
Rearranging equations is a fundamental skill when solving algebraic expressions. The goal is to simplify the equation to make it easier to solve. In our example, \(4x + 9 = 7x\), rearranging is done to get all terms with the variable \(x\) on one side and constants on the other.
A key strategy is to use inverse operations. Subtract \(4x\) from both sides to simplify the equation:
  • Left side: \(4x + 9 - 4x = 9\)
  • Right side: \(7x - 4x = 3x\)
This rearrangement isolates the variable term on one side, resulting in \(9 = 3x\).
Properly rearranging an equation makes the next steps, such as isolating variables, much clearer.
Isolating Variables
After rearranging the equation, the next step is to isolate the variable, which means getting the variable alone on one side of the equation. This process allows us to find the solution for the variable.
In \(9 = 3x\), the objective is to solve for \(x\). Since \(3x\) means 3 times \(x\), you need to divide both sides of the equation by 3 to isolate \(x\).
  • Divide the left side: \(9 \div 3 = 3\)
  • Divide the right side: \(3x \div 3 = x\)
You end up with \(x = 3\). Isolating variables often involves division, multiplication, addition, or subtraction.
These operations are applied equally to both sides of the equation to maintain balance and achieve the solution.

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Most popular questions from this chapter

Solve each inequality and check your solution. Then graph the solution on a number line. $$16-2 c<14$$

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CHALLENGE An apple costs the same as 2 oranges. Together, an orange and a banana cost \(10 \notin\) more than an apple. Two oranges cost \(15 \notin\) more than a banana. What is the cost for one of each fruit?
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