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Magazine Chapter 3: Problem 9
Solve the equations. $$ -4(2 m-5)=5 $$
Short Answer
Expert verified
Answer: The value of 'm' is 15/8.
Step by step solution
01
Distribute -4 to terms inside the parentheses
To distribute -4 to each term inside the parentheses, multiply -4 by each term:
$$
-4\cdot 2m + (-4)(-5) = 5
$$
02
Simplify the equation
After the distribution, we must simplify the equation by performing the multiplications:
$$
-8m + 20 = 5
$$
03
Move the constant to the right side of the equation
To move the constant 20 to the right side of the equation, subtract 20 from both sides:
$$
-8m = 5 - 20
$$
04
Simplify the equation
Simplify the equation by performing the subtraction on the right side:
$$
-8m = -15
$$
05
Divide by the coefficient of 'm'
Lastly, we need to find 'm' by dividing both sides of the equation by the coefficient of 'm' which is -8:
$$
m = \frac{-15}{-8}
$$
06
Simplify the fraction
Simplify the fraction to get the final value of 'm':
$$
m = \frac{15}{8}
$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Distributive Property
The distributive property is a handy tool for solving linear equations. It allows you to remove parentheses by distributing the multiplying factor across each term within the parentheses. For example, in our original problem: \(-4(2m - 5) = 5\), we apply the distributive property by multiplying \(-4\) with both \(2m\) and \(-5\). This gives us:
- \(-4 \cdot 2m = -8m\)
- \(-4 \cdot -5 = 20\)
- This results in the expression \(-8m + 20\), effectively removing the parentheses. By understanding this property, you can simplify complex equations more easily.
Simplifying Equations
Simplifying equations involves combining like terms and arranging the equation into a simpler form, making it easier to solve. After applying the distributive property, you often have to perform arithmetic operations to combine terms. In the example \(-8m + 20 = 5\), you might simplify by moving constants to one side:
- Subtract 20 from both sides to isolate terms with the variable \(-8m = 5 - 20\)
- \(-8m = -15\)
Isolating Variables
To solve an equation for a variable, we need to isolate it on one side of the equation. This means getting the variable term by itself. For the equation \(-8m = -15\), we need to find a way to express \(m\) in terms of known values. This is done by dividing both sides of the equation by the coefficient of \(m\), which is \(-8\):
- \(m = \frac{-15}{-8}\)
Fractions
Working with fractions is an integral part of algebra, particularly when finding solutions for equations. After isolating the variable, \(m\) in our example is represented as a fraction: \(\frac{-15}{-8}\). Not all fractions need simplification, but when possible, it can make the solution cleaner. Here:
- Dividing two negative numbers gives a positive result, so \(\frac{-15}{-8} = \frac{15}{8}\)
