Problem 25 - Prealgebra

Question

Solve each equation. Check your solution. $$-8 a=144$$

Step-by-Step Solution

Answer

The solution is \(a = -18\). Verified by substitution.

Step 1: Isolate the Variable

To solve the equation \(-8a = 144\), we begin by isolating the variable \(a\). This involves getting \(a\) by itself on one side of the equation. To do this, divide both sides of the equation by \(-8\). This gives us: \[ a = \frac{144}{-8} \]

Step 2: Simplify the Expression

Now that we have \(a = \frac{144}{-8}\), we simplify the expression. Divide 144 by -8 to find \(a\). This results in \(a = -18\), as \(144 \div 8 = 18\), and since one of the numbers is negative, the answer is negative.

Step 3: Verify the Solution

To verify that \(a = -18\) is the correct solution, substitute \(-18\) back into the original equation. Calculate \(-8 \times -18\) and confirm if it equals 144: \[ -8 \times -18 = 144 \] Since \(-8 \times -18 = 144\), our solution is correct.

Key Concepts

Isolation of Variables

The process of isolating the variable in an equation is an essential first step in solving linear equations. The goal is to manipulate the equation in such a way that the variable stands alone on one side, making it easy to solve for its value. In the equation \(-8a = 144\), we want to isolate \(a\), which involves undoing the multiplication by \(-8\). We do this by performing the inverse operation, which is division. By dividing both sides of the equation by \(-8\), the equation becomes clearer: \[a = \frac{144}{-8}\]. This simplification ensures that \(a\) is by itself on one side, ready for further simplification or verification. Remember, all operations must be performed equally on both sides to maintain the equation's balance.

Checking Solutions

Checking the solution to an equation is the final step to confirm its accuracy. Once you think you've isolated the variable and found its value, it's crucial to substitute this value back into the original equation to see if it satisfies the equation. For instance, if we found that \(a = -18\) in our equation \(-8a = 144\), we substitute \(-18\) for \(a\). Then, calculate \(-8 \times -18\).When this results in \(144\), as expected, our solution is correct.

• Substitute the variable with the found value.
• Perform the operations to see if the initial equality holds.
• If the original equation holds true, the solution is verified.

Verifying solutions not only supports your calculated answer but also strengthens understanding of the problem-solving procedure.

Simplifying Expressions

Simplifying expressions is a key skill in solving equations, as it makes working with equations and calculations easier. After isolating the variable, often the next step is to simplify the equation, so the solution becomes clearer.In the case of our example, the expression \(\frac{144}{-8}\) simplifies to \(-18\), accomplished by performing the division. Simplification involves operations such as:

• Dividing numbers to reduce fractions.
• Combining like terms when applicable.
• Using arithmetic operations to make calculations straightforward.

By simplifying expressions, you reduce the chance of computational errors, leading to more accurate solutions. Simplification helps in grasping the fundamental relationships within the equation, making it a fundamental part of math problem-solving.