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Chapter 2: Problem 7

Solve each equation, and check your solution. $$ x-12=19 $$

Short Answer

Expert verified
x = 31

Step by step solution

01

Isolate the variable

To solve for x, you need to isolate x on one side of the equation. Start by adding 12 to both sides of the equation:\[ x - 12 + 12 = 19 + 12 \]
02

Simplify both sides

Simplify both sides to simplify the equation. On the left side, \(-12 + 12\) cancels out, and on the right side, combine the constants:\[ x = 31 \]
03

Check your solution

Substitute the value you found for x back into the original equation to verify that it works:\[ 31 - 12 = 19 \]\[ 19 = 19 \] The solution is verified.

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Key Concepts

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

Isolate the Variable
When solving linear equations, isolating the variable is often the first essential step. In our example, the equation is \(x - 12 = 19\). To isolate the variable \(x\), you need to get it by itself on one side of the equation. Begin by performing the opposite operation to both sides of the equation. Here, we subtracted 12 from \(x\), so we need to add 12 to both sides to cancel out the -12. This gives us:

\[ x - 12 + 12 = 19 + 12 \]
After adding 12 on both sides, we simplify the equation to isolate \(x\).

This is your goal: ensuring that \(x\) (or whatever variable you are solving for) stands alone on one side of the equation.
Simplify Both Sides
Once you've taken steps to isolate the variable, the next task is to simplify both sides of the equation. This process clears the way to find the value of the variable. Our equation after isolating the variable was:

\[ x - 12 + 12 = 19 + 12 \]
Now, simplify the left side. Since \( -12 + 12 \) equals zero, we can remove it, leaving just \(x\):

\[ x = 19 + 12 \]
On the right side, add together the constants \(19 + 12 \), and you get:

\[ x = 31 \]
After these steps, we have simplified the equation to \(x = 31 \). This tells us what \(x\) is equal to.
Verify Solution
Verifying your solution is a crucial step to ensure that your calculations are correct. In this case, we found that \(x = 31 \). To check the solution, substitute \(31\) back into the original equation to see if it holds true:

\[ 31 - 12 = 19 \]
Simplify the left side:

\[ 19 = 19 \]
Since both sides of the equation are equal, the solution \(x = 31\) is verified as correct.

Always take this final step to double-check your work. It helps catch any errors and confirms your understanding of the problem.

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