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Chapter 4: Problem 12

Solve each equation. $$8(3 x-5)-4(2 x+3)=12$$

Short Answer

Expert verified
The solution to the given equation is \(x = 4\).

Step by step solution

01

Distribute

Apply the distributive property, which is \(a(b+c)=ab + ac\). So, distribute the 8 across the first parentheses \(8(3x - 5)\) to get \(24x - 40\) and the -4 across the second parentheses \(-4(2x + 3)\) to get \(-8x - 12\). The equation now reads \(24x - 40 - 8x - 12 = 12\).
02

Combine Like Terms

Combine terms on the left side of the equation to simplify. Combine \(24x - 8x\) to get \(16x\) and combine \(-40 - 12\) to get \(-52\). The equation now reads \(16x - 52 = 12\).
03

Isolate the variable

Add 52 to both sides of the equation to get \(16x = 64\). Then divide both sides by 16 to solve for \(x\). This gives \(x = 4\).

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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 cornerstone of algebra that allows us to simplify expressions and solve equations efficiently. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. In mathematical terms, it鈥檚 expressed as a(b + c) = ab + ac.

During the exercise, the distributive property is used to expand 8(3x - 5) and -4(2x + 3). This step transforms the equation from its compact form to a more workable version where like terms are visible and can be combined. This process is essential, as it sets the stage for further simplification and leads you towards the solution of the equation.
Combine Like Terms
Combining like terms is an integral part of simplifying algebraic expressions. Terms are 'like' if they have the same variable raised to the same power. The coefficient (the number in front of the variable) does not need to be the same. After distributing, we often find terms that can be combined to make the equation simpler.

In our example, 24x and -8x are like terms because they both contain the variable x to the first power. The same goes for -40 and -12 鈥� they are like terms, too, because they are constant numbers without any variable. By combining these, we simplify the left side of the equation, making it easier to isolate the variable in the next step.
Isolate the Variable
To find the value of the variable, we must isolate it on one side of the equation. It involves performing operations that 'undo' what is being done to the variable. In many cases, this means adding or subtracting terms to both sides to move constants, and multiplying or dividing to handle coefficients.

In our example, 16x - 52 = 12, we added 52 to both sides to remove the constant term from the left side. Following that, we divided both sides by 16, which is the coefficient of x. This process leaves us with x alone on one side and the solution, x = 4, on the other. Understanding how to isolate the variable is vital in solving equations, as it is the final step that uncovers the value we're seeking.

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