Problem 46 - Algebra 1: Concepts and Skills

Question

Check to see if the given value of the variable is or is not a solution of the equation. \(50=3 w ; w=15\)

Step-by-Step Solution

Answer

No, \( w = 15 \) is not a solution to the given equation.

Step 1: Understand the given equation and variable

The given equation is \( 50 = 3w \) and the value of variable \( w \) is provided as 15.

Step 2: Substitution

Replace \( w \) in the equation with the given value, yielding \(50 = 3 * 15\)

Step 3: Verification

Carry out the multiplication on the right side, which gives \( 50 = 45 \). On comparison, we see that these two numbers are not equal.

Key Concepts

Variable Substitution

Variable substitution is an essential concept in solving algebraic equations. It involves replacing a variable in an equation with a given or known value. This step allows us to transform an algebraic expression into an arithmetic one and make calculations simpler. For instance, in our original exercise, we substitute the variable \( w \) with 15 in the equation \( 50 = 3w \).

• Original equation: \( 50 = 3w \)
• Given value: \( w = 15 \)
• Substituted equation: \( 50 = 3 \times 15 \)

This substitution helps us to proceed by changing the problem from one involving a variable to one with only numbers. It's important to correctly substitute to ensure the outcome reflects the equation's relationships accurately.
If there are multiple variables, you'd substitute each one with its corresponding value, but in our case, there's just one, which simplifies the process considerably.

Equation Verification

Equation verification is the process of checking if the left side of the equation equals the right side after substitution. This step helps confirm whether the value substituted for the variable is indeed a solution.
The main goal is to see if the equation holds true. After substituting in our example, the equation becomes \( 50 = 3 \times 15 \), leading to the calculation of \( 3 \times 15 = 45 \). We then have the statement \( 50 = 45 \), which is clearly not true. This shows that the given value \( w = 15 \) is not a solution to the equation \( 50 = 3w \).

• If both sides of the equation do not match, then the substituted value is not a solution.
• If they match, then the substitution confirms the value is correct.

Always execute verification carefully to avoid errors in determining the truthfulness of the equation.

Multiplication in Equations

Multiplication is a fundamental arithmetic operation essential even in solving equations. In algebra, multiplication is often used to simplify equations or perform operations after variable substitution. When dealing with equations involving multiplication, it's vital to execute operations in the correct order.
In this exercise, once \( w \) is replaced by 15, the equation transforms into \( 50 = 3 \times 15 \).

• The multiplication \( 3 \times 15 \) results in 45.
• This demonstrates how operations are carried out, affecting the outcome of the verification process.

Properly carrying out multiplication ensures that any determined discrepancy, like in this case, can lead you to rethink your initial value or approach. In algebra, the accuracy in performing multiplication impacts the reliability of verifying equations and finding true solutions.