Problem 40 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The solution to the equation is x = 0.

Step 1: 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.\[-6(5x - 2) = (-6)(5x) + (-6)(-2) = -30x + 12\]​Now the equation becomes:\[8 + 4 = -30x + 12\]

Step 2: 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:\[12 = -30x + 12\]

Step 3: 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:\[0 = -30x\]

Step 4: 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.

Key Concepts

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.