Problem 11 - Algebra 1

Question

Check whether the given number is a solution of the equation. $$\frac{3}{4} x+1=-8 ;-12$$

Step-by-Step Solution

Answer

Yes, -12 is the solution of the given equation.

Step 1: Understand the Equation

The equation given is \(\frac{3}{4} x+1=-8\). We need to find out if when x = -12, this equation holds true.

Step 2: Substitute the provided value into the equation

Lets replace x with -12 in the equation which gives us \(\frac{3}{4}* -12 +1\). This simplifies to \( -9 +1 \).

Step 3: Compare the final result with the equation

The result is -8. The equation we were trying to solve also equates to -8. Hence, we can conclude -12 is indeed the solution to the equation.

Key Concepts

Substitution Method

When solving linear equations, the substitution method is a powerful tool. This method involves substituting a specific value into the equation to check if it satisfies the equation. In the given exercise, we are checking if \( x = -12 \) is a solution of the equation \( \frac{3}{4}x + 1 = -8 \). To apply the substitution method effectively:

• First, replace the variable in the equation, in this case, \( x \), with the provided number, which is \( -12 \).
• Make sure to carefully calculate the arithmetic operations. For our example, substitute \( -12 \) into the equation to get \( \frac{3}{4} \times -12 + 1 \).
• Simplifying this gives \( -9 + 1 = -8 \).

The calculated result should match the known solution from the original equation. If it does, as it does here, then the substituted value is indeed the correct solution.

Checking Solutions

After substituting a value into an equation, it's crucial to verify if the solution is correct. Checking solutions helps ensure that no mistakes were made in the substitution process or in the arithmetic calculations. In the case of the equation \( \frac{3}{4}x + 1 = -8 \), after substituting \( x = -12 \), we found our result as \( -8 \). To check the solution:

• Compare the simplified result from your substitution with the right side of the original equation. Here, \( -8 \) was expected and indeed was the outcome from our computation, confirming the correctness.
• If the numbers match, you can be confident that the solution is correct. If not, double-check your arithmetic steps and the substitution process.

This confirmation step is crucial as it reinforces the accuracy and reliability of your solution.

Fractional Coefficients

Handling fractional coefficients requires careful attention to detail. These coefficients, such as \( \frac{3}{4} \) in the equation \( \frac{3}{4}x + 1 = -8 \), affect the multiplication and simplification steps during substitution. Here's how to calmly maneuver through them:

• Multiply the fraction by the variable value carefully. For example, \( \frac{3}{4} \times -12 \) needs precise calculation.
• Breaking it down helps: multiply \( 3 \times -12 \) to get \( -36 \) and then divide by \( 4 \) to get \( -9 \).
• Fractions can often be simplified, so always look out for potential simplifications to make calculations easier.

Proper management of fractional coefficients ensures accuracy in solving equations, avoiding common pitfalls associated with partial calculations.