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Chapter 2: Problem 49

Solve each equation. $$8(4 x+5)-5(6 x)-x=53-6(x+1)+3(2 x+2)$$

Short Answer

Expert verified
x = 7

Step by step solution

01

Distribute

Apply the distributive property to both sides of the equation. For the left side, distribute 8 to both terms inside the parenthesis and -5 to the term inside parenthesis. For the right side, distribute -6 and 3 to their respective terms.\[ 8(4x + 5) - 5(6x) - x = 53 - 6(x + 1) + 3(2x + 2) \]This becomes:\[ 32x + 40 - 30x - x = 53 - 6x - 6 + 6x + 6 \]
02

Combine Like Terms

Simplify both sides by combining like terms.Left side:\[ 32x - 30x - x + 40 = 32x - 31x + 40 = x + 40 \]Right side:\[ 53 - 6 + 6x - 6x + 6 = 47 \]So, the equation simplifies to:\[ x + 40 = 47 \]
03

Solve for x

Isolate the variable x by subtracting 40 from both sides of the equation.\[ x + 40 - 40 = 47 - 40 \]\[ x = 7 \]

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Key Concepts

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

distributive property in algebra
The distributive property is a key concept in algebra that helps simplify expressions. It allows you to multiply a single term by each term inside a parenthesis. This way, you can eliminate the parentheses and work with simpler expressions. For example, in the equation \[ 8(4x + 5) - 5(6x) - x \ = 53 - 6(x + 1) + 3(2x + 2) \], you need to distribute the coefficients 8, -5, -6, and 3 to the respective terms inside the paranthesis. When you apply the distributive property, you get: \[ 32x + 40 - 30x - x = 53 - 6x - 6 + 6x + 6 \]. By distributing, we can now see each term separately and prepare to combine like terms.
combining like terms
Combining like terms is another essential step in solving linear equations. This means gathering similar terms, terms that have the same variables raised to the same power. In our example, after distribution we have: \[32x + 40 - 30x - x = 53 - 6x - 6 + 6x + 6 \] On the left side, the like terms are the terms that include 'x': 32x, -30x, and -x. Simplifying these, we get:ewline \[ 32x - 30x - x + 40 \] This becomes: \[ x + 40 \]On the right side, we need to combine all constant terms and terms involving 'x': ewline 53 - 6 + 6x - 6x + 6, by simplifying these we get:ewline \[53 - 6 + 6 = 47 \]. Now, our equation is ready for solving: ewline \[x + 40 = 47 \]
isolating the variable
Isolating the variable means getting the variable, in this case 'x', alone on one side of the equation. This will help you find the value of the variable. From our simplified equation: ewline \[ x + 40 = 47 \] you need to eliminate the constant on the left by its inverse operation. Subtract 40 from both sides: ewline \[ x + 40 - 40 = 47 - 40 \] which simplifies to: ewline \[ x = 7 \] Thus, through isolating the variable, you have determined that ewline \[ x = 7 \] By applying the distributive property, combining like terms, and isolating the variable, you can solve linear equations effectively.

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