Problem 22 - Algebra 1

Question

CHECKING SOLUTIONS OF EQUATIONS Check whether the given number is a solution of the equation. $$m+4 m=60-2 m ; 10$$

Step-by-Step Solution

Answer

No, 10 is not a solution to the equation.

Step 1: Substitute the value

Substitute \(m\) with 10 in the equation \(m + 4m = 60 - 2m\). This gives us \(10 + 4(10) = 60 - 2(10)\).

Step 2: Solve both sides of the equation

Solve the left-hand side and the right-hand side of the equation. The left-hand side simplifies to \(10 + 40 = 50\), and the right-hand side simplifies to \(60 - 20 = 40\).

Step 3: Evaluate the two sides of the equation

Compare the results from both sides to see if they equate. Since 50 ≠ 40, the two sides of the equation are not equal. Therefore, the number 10 is not a solution to the equation.

Key Concepts

Substituting Values

Understanding how to check solutions of equations by substituting values is an essential skill in algebra. Substitution involves replacing a variable, which in this case is represented by a letter such as 'm', with a specific numerical value. For example, given the equation m + 4m = 60 - 2m, we can verify if a number is a solution by following a simple process.

The first step is to take the proposed solution, let's say '10', and substitute it into the equation wherever 'm' appears, yielding 10 + 4(10) = 60 - 2(10). This allows us to see if the equation holds true for that particular value. When students tackle this step, it's crucial to ensure they replace every instance of the variable and apply proper arithmetic operations to avoid any mistake.

Solving Equations

Solving equations is about finding the values for the variables that make the equation true. After substituting a given value into an equation, you ascertain if the substitution leads to a true statement. To do this, perform the necessary calculations on both sides. In our example, after replacing 'm' with '10', you would solve the left-hand side as 10 + 40 = 50, and the right-hand side as 60 - 20 = 40.

This process includes simplifying the equation by executing basic arithmetic — addition, subtraction, multiplication, and division. It’s vital for students to handle this step with diligence to ensure each operation adheres to the correct order.

Evaluate Equations

Once you've solved both sides of the equation post-substitution, the final step is to evaluate if these expressions equate. This step essentially means comparing the simplified forms of the left and right-hand sides to see if they match. In the case of our example, we compare '50' from the left to '40' from the right. If they are equal, the substituted value is a solution to the equation.

However, as seen in the example, since '50' does not equal '40', the statement is false, indicating that '10' is not a solution. Evaluating the equations verifies the validity of a proposed solution. It's significant for students to comprehend that a solution to the equation exists only when these final evaluated sides are identical.