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Chapter 2: Problem 26

Solve each equation. Check all solutions. SEE EXAMPLE 1. (OBJECTIVE 1) $$7 x-19=2$$

Short Answer

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

Step by step solution

01

Isolate the x term

The first step is to isolate the term with \(x\). To do that, add 19 to both sides of the equation to get rid of the -19 on the left side. The equation becomes: \(7x = 2 + 19\).
02

Simplify the Equation

After carrying out the calculation on the right side of the equation from step 1, it simplifies to \(7x = 21\).
03

Solve for x

The final step in solving for \(x\) is to divide each side of the equation by 7. This gives: \(x = 21 / 7\).
04

Check the Solution

To check the solution, substitute \(x = 3\) back into the original equation \(7x - 19 = 2\) and make sure both sides of the equation are equal.

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Key Concepts

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

Isolate the Variable
When you're faced with a linear equation like \(7x - 19 = 2\), the ultimate goal is to figure out 'what x equals'. This process starts with isolating the variable, which means getting the variable on one side of the equal sign and everything else on the other. Think of it as giving 'x' its own space to shine without any distractions.

To begin, adjust the equation by adding or subtracting terms to move them away from the variable. It鈥檚 like telling the number 19 to hop over to the other side of the seesaw so that 'x' can sit alone, which in this case means adding 19 to both sides. You're left with a much cleaner \(7x = 21\), setting the stage for the next crucial steps.
Simplify the Equation
Once you've given 'x' its own territory, the next goal is to simplify the equation, which is essentially tidying up. It's kind of like when you clear off a cluttered desk鈥攕uddenly it鈥檚 easier to work. In our equation, we started with \(7x = 2 + 19\). To simplify, we combine like terms, which in this case means doing the simple arithmetic of adding 2 and 19 to get \(7x = 21\).

After simplifying, it often becomes clear what to do next. Just like having a clean desk helps you focus on homework, a simplified equation helps you see the solution clearly: now we can easily spot that we need to divide both sides by 7 to find the value of 'x'.
Check the Solution
After doing the math to find out that \(x = 3\), one might think the mission is accomplished. However, to seal the deal, it's vital to check the solution. This is like retracing your steps to ensure you haven鈥檛 dropped your keys somewhere. Plugging \(x = 3\) back into the original equation gives us \(7(3) - 19\), which simplifies to \(21 - 19 = 2\), mirroring the original equation.

By verifying that both sides equal each other, confidence in the solution skyrockets! It's the final handshake to confirm that yes, indeed, \(x = 3\) is the correct answer, and you've solved the linear equation like a pro.

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