Problem 22 - Prealgebra

Question

Solve each equation. Check your solution. $$\frac{m}{45}=-3$$

Step-by-Step Solution

Answer

The solution to the equation is \( m = -135 \).

Step 1: Isolate the Variable

To solve the equation \( \frac{m}{45} = -3 \), we need to isolate \( m \). To do this, multiply both sides of the equation by 45 to cancel out the denominator. This yields: \[ m = -3 \times 45 \]

Step 2: Calculate the Product

Now, perform the multiplication on the right side of the equation: \[ m = -3 \times 45 = -135 \]

Step 3: Check the Solution

To ensure that \( m = -135 \) is the correct solution, substitute \( m = -135 \) back into the original equation: \[ \frac{-135}{45} = -3 \] Perform the division:\[ -135 \div 45 = -3 \] This confirms that the original equation holds true, so \( m = -135 \) is indeed the correct solution.

Key Concepts

Solving Equations

Solving equations is an essential part of algebra, where we are tasked with finding the value of an unknown variable that makes the equation true. In this case, we have the equation \( \frac{m}{45} = -3 \). To solve it, we must "isolate" the variable \( m \). This means we need to get \( m \) by itself on one side of the equation. To achieve this, think about the opposite operations. The equation currently divides \( m \) by 45. Therefore, we need to do the opposite of division, which is multiplication, to get rid of the 45.

• Multiply both sides by 45
• This removes the fraction
• Resulting equation: \( m = -3 \times 45 \)

By multiplying both sides by the same number, we keep the equation balanced and correctly isolate \( m \). After following these steps, you will find the value of \( m \) that satisfies the equation.

Arithmetic Operations

Arithmetic operations are the basic calculations we perform such as addition, subtraction, multiplication, and division. In solving algebraic equations, these operations help us manipulate the equations to isolate the variable.In our current problem, once we isolate \( m \) by multiplying both sides by 45, we then need to perform the arithmetic operation of multiplication on the right side: \( -3 \times 45 \).

• Calculate \( -3 \times 45 \)
• Think of multiplication as repeated addition
• \( -3 \times 45 = -135 \)

The result of this computation gives us \( m = -135 \). Understanding these operations is key to efficiently solving equations. Focusing on one operation at a time ensures clarity and accuracy.

Checking Solutions

After finding a solution to an equation, checking your work is crucial. It's like reviewing your paper before submitting it. Once we find \( m = -135 \), we substitute it back into the original equation to verify its correctness.

• Substitute \( m = -135 \) into \( \frac{m}{45} \)
• Calculate \( \frac{-135}{45} \)
• Ensure it equals \(-3\)

By performing this division, we see \(-135 \div 45 = -3\). The equation holds true, confirming that our solution is correct. This step is essential in algebra to ensure no errors were made during solving. Always double-check!