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Chapter 15: Problem 26

Solve. $$ 2(5-m)=5(6+m) $$

Short Answer

Expert verified
The solution to the equation is $$ m = -\frac{20}{7} $$

Step by step solution

01

- Distribute

Distribute the numbers outside the parentheses. Multiply 2 by both 5 and -m on the left side, and 5 by both 6 and m on the right side.This gives us: $$ 2 \times 5 - 2 \times m = 5 \times 6 + 5 \times m$$ Which simplifies to:$$ 10 - 2m = 30 + 5m$$
02

- Combine Like Terms

Move all terms involving m to one side of the equation and constant terms to the other side. Add 2m to both sides and subtract 30 from both sides: $$ 10 - 30 = 30 - 30 + 5m + 2m $$ Which simplifies to: $$ -20 = 7m $$
03

- Solve for m

Divide both sides by 7 to solve for m: $$ \frac{-20}{7} = m $$ So, the simplified form is: $$ m = -\frac{20}{7} $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distribute Property
The distributive property is a vital concept in solving linear equations. It allows us to multiply a single term by each term inside a set of parentheses. In our exercise, we started with the equation: $$ 2(5-m) = 5(6+m) $$ To apply the distributive property, we multiply 2 by both 5 and -m on the left side, and 5 by both 6 and m on the right side: $$ 2 \times 5 - 2 \times m = 5 \times 6 + 5 \times m $$ Simplifying these expressions gives us: $$ 10 - 2m = 30 + 5m $$ By using the distributive property, we convert the equation into a simpler form that we can solve more easily.
Combine Like Terms
Combining like terms helps to simplify an equation by bringing similar terms together. Like terms are terms that have the same variables raised to the same power. In our simplified equation from the previous step: $$ 10 - 2m = 30 + 5m $$ We need to get all the terms with the variable 'm' on one side and the constant numbers on the other side. To do this, we add 2m to both sides and subtract 30 from both sides: $$ 10 - 30 = 30 + 5m + 2m - 30 $$ This simplifies to: $$ -20 = 7m $$ When the like terms are combined, the equation becomes much simpler to solve.
Isolating the Variable
Isolating the variable means getting the variable 'm' alone on one side of the equation to solve for its value. In the current simplified equation: $$ -20 = 7m $$ We need to isolate 'm' by dividing both sides by 7: $$ \frac{-20}{7} = m $$ This simplifies to: $$ m = -\frac{20}{7} $$ By isolating the variable, we have found the value of 'm'. This step is crucial because it provides the solution to the linear equation. Always remember to check your work to ensure there are no mistakes in your calculations.

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