Problem 14 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation using the multiplication property of equality. Be sure to check your proposed solutions. $$-8 x=4$$

Step-by-Step Solution

Answer

The solution to the equation \(-8 x=4\) is \(x = -0.5\).

Step 1: Isolate the variable \(x\)

Start by isolating the variable \(x\) on one side of the equation. You can achieve this by dividing both sides of the equation by \(-8\). The equation then becomes \(x = -\frac{4}{8} = -0.5\)

Step 2: Check the proposed solution

Substitute the value for \(x\) which we found as \(-0.5\) back into the original equation, which is \(-8 x=4\). This gives \(-8(-0.5) = 4\), which simplifies to \(4 = 4\). Since both sides of the equation are equal, this means that \(x = -0.5\) is indeed the correct solution for the equation \(-8 x=4\).

Key Concepts

Understanding the Multiplication Property of Equality

The multiplication property of equality is a fundamental principle used to solve linear equations. It states that if you multiply both sides of an equation by the same nonzero number, the equality is still true. For example, if you have an equation like \(2x = 6\), you can multiply both sides by \(\frac{1}{2}\) to isolate \(x\), yielding \(x = 3\).

It's crucial to remember that whatever you do to one side of the equation, you must also do to the other side to maintain balance. In our exercise, dividing by \( -8 \) is applying the multiplication property of equality because division is the same as multiplying by a reciprocal. This step changes the equation from \( -8x = 4 \) to \( x = -0.5 \) without disrupting the balance of the equation.

Isolating the Variable

The goal when isolating the variable in an algebraic equation is to get the variable on one side of the equation by itself. This process involves performing operations that 'undo' what is being done to the variable. For the equation \( -8x = 4 \), the variable \(x\) is being multiplied by \( -8 \). To isolate \(x\), you'd need to do the opposite of multiplication, which is division in this case.

To accomplish this, you divide both sides by \( -8 \), which effectively cancels out the \( -8 \) on the left and leaves \(x\) by itself. It's a critical step in solving equations because it allows you to find the value of the variable.

Checking Solutions in Algebra

After finding a solution to an algebraic equation, it's important to check the solution to ensure it's correct. This is done by substituting the value back into the original equation and verifying that both sides remain equal.

In our example, after finding \(x = -0.5\), we check by calculating \( -8 (-0.5) \) to see if it indeed equals \( 4 \). Since it does, the solution is confirmed. This step is not just about verifying your answer, but also understanding the equation better and gaining confidence in your algebraic skills.