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Chapter 2: Problem 19

For exercises 1-72, (a) solve. (b) check. $$ \frac{1}{2} x-10=-24 $$

Short Answer

Expert verified
The solution to the equation is \( x = -28 \). This solution checks out correctly.

Step by step solution

01

- Move the constant term

Add 10 to both sides of the equation to isolate the term with the variable. \[ \frac{1}{2}x - 10 + 10 = -24 + 10 \] Simplifying this, we get \[ \frac{1}{2}x = -14 \]
02

- Solve for x

To isolate x, multiply both sides of the equation by 2. \[ 2 \times \frac{1}{2}x = -14 \times 2 \] Simplifying, this gives \[ x = -28 \]
03

- Check the solution

Substitute \( x = -28 \) back into the original equation to verify it satisfies the equation. \[ \frac{1}{2}(-28) - 10 = -24 \] Calculate: \[ -14 - 10 = -24 \] Since both sides of the equation are equal, the solution \( x = -28 \) is correct.

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Key Concepts

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

Isolating Variables
Isolating variables is the key to solving equations. The goal is to get the variable alone on one side of the equation. For example, if we start with the equation:
\(\frac{1}{2}x - 10 = -24\), our first step is to move the constant term to the other side. We add 10 to both sides to achieve this:
\(\frac{1}{2}x - 10 + 10 = -24 + 10\).
By simplifying, we get:
\(\frac{1}{2}x = -14\)
Now, the variable term is isolated, leaving us with a simpler equation to solve.
Remember, whatever you do to one side of the equation, you must do to the other side to maintain balance. This is the fundamental principle of isolating variables.
Checking Solutions
After solving for the variable, it鈥檚 crucial to check the solution. Substitute the value back into the original equation to ensure it holds true. In our example, the solution we found was \x = -28\.
Let's substitute this value back into the original equation: \(\frac{1}{2}(-28) - 10 = -24\).
Calculate step-by-step:
\(\frac{1}{2}(-28) = -14\)
Then, \-14 - 10 = -24\
Both sides of the equation are equal, confirming that \x = -28\ is indeed the correct solution.
Checking solutions may seem like an extra step, but it's essential to ensure accuracy.
Multiplication and Division of Equations
Multiplication and division play a crucial role in solving equations, especially when you need to clear fractions or coefficients. For example, after isolating the variable term in the equation to:
\(\frac{1}{2}x = -14\), the next step is to get \x\ by itself. We can multiply both sides by 2 to eliminate the fraction:
\2 \times \frac{1}{2}x = -14 \times 2\
Simplifying this equation, we get: \x = -28\
Multiplying or dividing both sides of an equation by the same number keeps the equation balanced. It鈥檚 important to perform these operations correctly to isolate the variable effectively.
Simplifying Expressions
Simplifying expressions makes equations easier to solve. This involves combining like terms and performing basic arithmetic operations. In our example, after moving the constant term:
\(\frac{1}{2}x - 10 + 10 = -24 + 10\), we simply calculate:
\(\frac{1}{2}x = -14\)
Each step of simplification brings us closer to isolating the variable. Furthermore, while checking solutions, simplification ensures that calculations are manageable and reduces the risk of errors. Always simplify where possible before moving on to the next step.

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Most popular questions from this chapter

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