Problem 16 - Algebra 1: Concepts and Skills

Question

Check to see if \(a=5\) is or is not a solution of the equation. $$ \frac{40}{a}=8 $$

Step-by-Step Solution

Answer

Yes, \(a=5\) is a solution of the equation \(\frac{40}{a}=8\).

Step 1: Substitute the Value

Place the value of \(a=5\) into the equation \(\frac{40}{a}=8\). This provides us with the new equation: \(\frac{40}{5}=8\)

Step 2: Simplify the Equation

Simplify the left hand side of the equation by performing the division. This simplifies to: \(8=8\).

Step 3: Verify Equality

Now as we see that after the substitution both sides of the equation are equal. Hence, \(a=5\) is a valid solution for the given equation.

Key Concepts

Substitution Method

The substitution method is a fundamental technique used in solving equations. It involves replacing a variable with a given value to test if it satisfies the original equation. In many cases, like our exercise, we begin with an equation and a potential solution for a variable, such as \(a=5\). To apply substitution, follow these steps:

• Identify the variable to replace, here it's \(a\).
• Substitute the variable with the given value in the equation. In our example, this transforms \(\frac{40}{a}=8\) into \(\frac{40}{5}=8\).
• Simplify any calculations involved to see if both sides of the equation remain equal.

Substitution is especially useful in systems of equations where multiple variables are present. By replacing one variable and solving for the others, you can determine if the proposed values work across all equations.

Verification of Solution

Once the substitution is done, the next step in solving equations effectively is to verify the solution. Verification checks: Does the substituted value make the equation a true statement?

• After substituting \(a=5\) in our equation \(\frac{40}{a}=8\), we simplify it to find \(\frac{40}{5}=8\).
• Simplifying further gives us \(8=8\), a true statement, confirming our solution.

Verification is crucial because it ensures accuracy. Without it, there is a risk of proceeding with an incorrect solution, especially in complex equations. This step guarantees that both sides of the initially given equation are equal when the potential solution substitutes the variable.

Division in Equations

Division in equations is a crucial mathematical operation that needs careful handling. It involves isolating one part of an equation through division to simplify and solve it.In our example, the equation \(\frac{40}{a}=8\) inherently involves division. Here are some important aspects to consider:

• Perform the division only on the variable side to simplify and check equations.
• Ensure that the divisor is not zero, as division by zero is undefined in mathematics.
• In our equation, \(40\) is divided by \(5\) to check if the solution is correct, resulting in \(8\). Both sides match, validating our solution.

Understanding how division works within equations is essential for simplifying and verifying solutions. It aids in isolating variables and validating potential solutions, ensuring they make both mathematical and logical sense.