Problem 39 - Algebra 1: Concepts and Skills

Question

SOLVING EQUATIONS Multiply by a reciprocal to solve the equation. $$ 0=\frac{7}{8} x $$

Step-by-Step Solution

Answer

0 is the solution to the equation.

Step 1: Understand the initial equation

The initial equation is 0 = \(\frac{7}{8} x\). The goal is to get x by itself on one side of the equation.

Step 2: Isolate x

To isolate x, multiply both sides of the equation by the reciprocal of \(\frac{7}{8}\), which is \(\frac{8}{7}\). However, multiplying zero by any number still results in zero. So, we have 0 = x.

Step 3: Re-check the equation

Substitute 0 back into the original equation and check. So, \(0 = \frac{7}{8}*0 = 0\). 0 does indeed equal 0, meaning the solution is valid.

Key Concepts

Multiplying by Reciprocals

When solving equations, one effective method is to multiply by the reciprocal. A reciprocal is basically flipping the numerator and the denominator of a fraction. For a fraction like \( \frac{7}{8} \), its reciprocal would be \( \frac{8}{7} \). Multiplying by a reciprocal is especially useful because it helps us eliminate fractions in equations to simplify and solve them.

In the exercise, you have the equation \( 0 = \frac{7}{8} x \). Our aim is to solve for \( x \). By multiplying both sides of the equation by the reciprocal of \( \frac{7}{8} \), which is \( \frac{8}{7} \), the fraction on the right side cancels out.

• Multiply \( \frac{8}{7} \times \frac{7}{8} x \)
• This simplifies to \( 1x \) or just \( x \)

The left side remains \( 0 \times \frac{8}{7} \), which is still 0. This operation is crucial in isolating the variable, thereby simplifying our problem.

Isolating Variables

Isolating a variable is at the heart of solving algebraic equations. It means rearranging the equation until the variable you are solving for stands alone on one side. In our problem \( 0 = \frac{7}{8} x \), we need to get \( x \) all by itself.

After multiplying by the reciprocal in the previous step, you are effectively working to cancel the fraction coefficient from \( x \). This process is essential because:

• It simplifies your equation to easier numbers or even whole numbers.
• It gives a clearer picture of the value of the variable.

Remember that an equation is like a balance scale. Whatever you do to one side, you must do to the other. Hence, multiplying both sides by \( \frac{8}{7} \) ensures the equation remains balanced, leading you to \( 0 = x \). This step-by-step strategy ultimately isolates \( x \), making the equation much easier to interpret and solve.

Validating Solutions

Validating solutions involves checking whether the solution we have obtained meets the requirements of the original equation. It's the last but vital step to ensure everything was done correctly.

Once you have isolated the variable and found \( x = 0 \), it's essential to substitute it back into the original equation to verify correctness:

• Substitute \( x = 0 \) into \( \frac{7}{8} x \)
• This gives \( \frac{7}{8} \times 0 = 0 \)
• Check that this matches the left side of the original equation, which is also 0.

This step might seem small, but it assures you that the calculations were accurate. By re-affirming the equality, you confirm that no mistakes were made along the way, and the solution satisfies the equation, thereby providing confidence in your solution.