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Chapter 2: Problem 40

Solve each equation. See Examples 9 and \(10 .\) \(8+4=-6(5 x-2)\)

Short Answer

Expert verified
The solution to the equation is x = 0.

Step by step solution

01

Simplify the Right Side of the Equation

Start by distributing the -6 into the expression (5x - 2) on the right side. This means you will multiply -6 by both 5x and -2.\[-6(5x - 2) = (-6)(5x) + (-6)(-2) = -30x + 12\]鈥婲ow the equation becomes:\[8 + 4 = -30x + 12\]
02

Simplify the Left Side of the Equation

Combine like terms on the left side of the equation. You have two constants:\[8 + 4 = 12\]Therefore, the equation simplifies to:\[12 = -30x + 12\]
03

Isolate the Variable Term

Subtract 12 from both sides to move the constant term on the right side:\[12 - 12 = -30x + 12 - 12\]This simplifies to:\[0 = -30x\]
04

Solve for the Variable

Divide both sides of the equation by -30 to solve for x:\[\frac{0}{-30} = \frac{-30x}{-30}\]Therefore, x = 0.

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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 fundamental concept in algebra that helps us simplify expressions and solve equations. It involves distributing a multiplier to each term within a parenthesis. In simpler terms, if you have an expression like \(a(b + c)\), the distributive property allows us to write it as \(ab + ac\).

In our exercise, we use the distributive property on the expression \(-6(5x - 2)\). Here's what happens:
  • Multiply \(-6\) by \(5x\) to get \(-30x\).
  • Then, multiply \(-6\) by \(-2\) to get \(12\).
So, the expression simplifies from \(-6(5x - 2)\) to \(-30x + 12\).

Using the distributive property not only helps in simplifying complex expressions but also sets the stage for solving equations efficiently.
Combining Like Terms
Combining like terms is a key algebraic skill used to simplify expressions and equations. Like terms are terms that have the same variable raised to the same power. When you combine them, you simply add or subtract their coefficients.

In the exercise, we see this under the Left Side of the equation:
  • Original equation: \(8 + 4 = -30x + 12\)
  • Combine the constants on the left: \(8 + 4\) becomes \(12\).
This results in the simplified equation: \(12 = -30x + 12\).

By combining like terms, the equation becomes much more manageable, allowing us to focus on isolating the variable to find its value.
Solving Linear Equations
Solving linear equations is a central concept in algebra that involves finding the value of the variable that makes the equation true. Linear equations are equations of the first degree, which means they have no exponents larger than one.

Here's how we solve the linear equation \(12 = -30x + 12\) from our exercise:
  • Subtract \(12\) from both sides to isolate the term with \(x\): \(12 - 12 = -30x + 12 - 12\) simplifies to \(0 = -30x\).
  • Divide both sides by \(-30\) to solve for \(x\): \(\frac{0}{-30} = \frac{-30x}{-30}\), which simplifies to \(x = 0\).
By following these steps, we isolate the variable and determine that \(x = 0\) is the solution to the equation. This process highlights that once like terms are simplified, solving linear equations becomes a straightforward task of isolating and finding the value of the variable.

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Most popular questions from this chapter

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