Problem 89 - Algebra A Combined Function

Question

Fill in the blank with a number of your choice so that each equation has the given solution. 6 x= ____ ; solution: -8

Step-by-Step Solution

Answer

The blank should be filled with -48.

Step 1: Identify Given Equation Structure

We are given the equation of the form \(6x = \text{blank}\) and it needs to have \(x = -8\) as the solution.

Step 2: Substitute Known Solution into Equation

Substitute the given value of \(x\) into the left-hand side of the equation. This means we will calculate \(6 \times (-8)\).

Step 3: Perform the Multiplication

Carry out the multiplication: \(6 \times (-8) = -48\). This result will fill the blank in the equation.

Step 4: Write the Completed Equation

After substitution and calculation, the equation becomes \(6x = -48\). This confirms the solution \(x = -8\).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of unknown variables that make an equation true. In our example, we started with the equation form \(6x = \text{blank}\) and needed to confirm that \(x = -8\) is the correct solution. Solving the equation involves replacing the variable with the given solution and checking if both sides of the equation are equal. Here, we substitute \(-8\) for \(x\) and then calculate the result to see if it fills the blank correctly. This process of substitution and calculation ensures that the solution is correct and satisfies the equation.

Multiplication in Algebra

Multiplication in algebra is similar to multiplication in basic arithmetic but involves variables and constants. When we multiply the number 6 by the variable \(x\), it represents scaling the variable by 6 times. In our case, we needed to find the product of 6 and \(-8\).

• Step 1: Write the expression \(6 \times (-8)\).
• Step 2: Calculate the product, which is \(-48\).

The product \(-48\) is the result of multiplying these numbers together, which fills our blank. Multiplication in algebra helps us understand how variables interact with constants to form complete expressions and equations.

Substitution Method

The substitution method is an important strategy in algebra used to solve equations by replacing variables with their known values. For the equation \(6x = ?\), we were given the solution \(x = -8\). We employed substitution to determine what the equation's "blank" should be by

• Substituting \(-8\) for \(x\).
• Calculating the expression \(6 \times (-8)\).
• Verifying the solution as \(-48\).

By using substitution, we can evaluate algebraic expressions with specific values, providing a concrete method for checking solutions and filling gaps in equations. This method is particularly useful when you have one variable and know its value.

Negative Numbers in Algebra

Working with negative numbers in algebra requires careful attention to the rules of arithmetic, especially regarding multiplication. In the equation \(6x = ?\), we multiplied a positive number by a negative one. Here is what to keep in mind:

• Multiplying two positive numbers gives a positive result.
• Multiplying a positive by a negative number gives a negative result.
• Multiplying two negative numbers results in a positive number.

Thus, 6 multiplied by \(-8\) gives \(-48\). These rules can be tricky, but understanding them is essential for correctly solving algebraic equations. Negative numbers can indicate direction or inverse relationships in mathematical contexts, providing depth to algebraic solutions.