Problem 44 - Algebra 1

Question

Solve the equation. Round the result to the nearest tenth if necessary. $$12 m-9=5 m-2$$

Step-by-Step Solution

Answer

The value of 'm' is approximately 1.6, when rounded to the nearest tenth

Step 1: Combine like terms

Given equation is \(12 m-9=5 m-2\). First, we need to gather the 'm' terms on one side and the numbers on the other. You can do this by subtracting '5m' from both sides, giving the equation \(12m - 5m = 2 + 9\)

Step 2: Simplification

This simplifies to \(7m = 11\).

Step 3: Solve for 'm'

Now, isolate 'm' by dividing the entire equation by 7. The equation becomes \(m = 11 / 7\).

Key Concepts

Equation Solving

Solving equations is like detective work. You have all these clues, or terms, sitting in front of you and need to figure out how they fit together. When you are faced with an equation like \(12m - 9 = 5m - 2\), the goal is to make both sides equal by moving things around.
This usually involves shifting terms, combining them, and working towards seeing the connection between different variables and numbers.
In our exercise, you start with locating all the 'm' terms and the number terms to combine and simplify them. This is where your understanding of like terms comes in and helps solve the equation in a systematic manner. Equation solving basically asks you to find that missing number that makes the entire statement true. It's all about balance!

• Keep both sides of the equation balanced as you work through them.
• Apply arithmetic operations equally to ensure the equation stays correct.

Like Terms

Like terms are like peas of the same pod. They are terms that contain the same variables raised to the same power. These terms can be directly added together to simplify an equation.
In the given equation, we have terms \(12m\) and \(5m\). These are like terms since both terms have the variable 'm'.
Combining them means we perform the operation necessary, such as subtraction in this case, to simplify calculations. When we subtract \(5m\) from \(12m\), we end up with \(7m\). This now makes the next steps towards solving the equation straightforward and simpler.

• Always gather and combine like terms to ease calculations.
• These terms allow for straightforward addition or subtraction without changing the equation's meaning.

Variable Isolation

Variable isolation is about "freeing" the variable. It is the central focus when solving linear equations. In our exercise, the equation simplifies down to \(7m = 11\). Here, 'm' still feels a bit trapped by 7.
We need to set 'm' free to find its value. To do this, you divide both sides by the coefficient that is attached to 'm', which, in this case, is 7.
This step yields \(m = 11/7\), completing the equation solution. This simple step of isolating the variable leads you to the value that solves the original equation.

• Simplify the equation such that only the variable remains on one side.
• Perform any arithmetic operations necessary to "untangle" the variable from other numbers.
• This isolation gives you the exact value of the variable.