Problem 51 - Intermediate Algebra

Question

Solve each equation. Check each result. See Example 5. $$ 9(x-2)=-6(4-x)+18 $$

Step-by-Step Solution

Answer

The solution is \( x = 4 \).

Step 1: Distribute on both sides of the equation

Distribute the numbers outside the parentheses on both sides of the equation. Multiply 9 by each term inside the parentheses on the left side and -6 by each term on the right side. This gives: \[ 9(x - 2) = 9x - 18 \]\[ -6(4 - x) = -24 + 6x \] so the equation becomes:\[ 9x - 18 = -24 + 6x + 18 \]

Step 2: Simplify the equation

First, simplify the right side of the equation by adding -24 and 18:\[ -24 + 18 = -6 \]Now we have:\[ 9x - 18 = 6x - 6 \]

Step 3: Collect like terms

To isolate terms containing \( x \), subtract \( 6x \) from both sides:\[ 9x - 6x - 18 = -6 \]This simplifies to:\[ 3x - 18 = -6 \]

Step 4: Solve for x

Add 18 to both sides to isolate the \( x \)-term:\[ 3x - 18 + 18 = -6 + 18 \]Which simplifies to:\[ 3x = 12 \]Now divide both sides by 3 to solve for \( x \):\[ x = 4 \]

Step 5: Check the solution

To verify, substitute \( x = 4 \) back into the original equation: \[ 9(4 - 2) = -6(4 - 4) + 18 \] Simplify both sides: \[ 18 = 0 + 18 \] Since both sides equal, the solution \( x = 4 \) is correct.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions. When you see an expression like \( a(b + c) \), the distributive property allows you to distribute \( a \) to both \( b \) and \( c \). This is done by multiplying \( a \) by each term within the parentheses. In symbolic terms, it's written as:

• \( a(b + c) = ab + ac \)

In the given equation, we apply this property to both sides:

• Left side: \( 9(x - 2) \) becomes \( 9x - 18 \).
• Right side: \( -6(4 - x) \) turns into \( -24 + 6x \).

This property is particularly helpful in linear equations as it helps remove parentheses and simplifies the expression, which is a crucial first step in solving such equations.

Combining Like Terms

Once the distributive property has been applied and parentheses are eliminated, the next step in solving linear equations is to combine like terms. Like terms are terms that contain the same variable raised to the same power. For example, \( 3x \) and \( 5x \) are like terms, but \( 3x \) and \( 5y \) are not because they contain different variables.In this exercise, after using the distributive property, the equation becomes:

• \( 9x - 18 = 6x + (-6) \)

Here, you combine like terms on the left side and the right side of the equation:

• On the left side, there are no other \( x \)-terms apart from \( 9x \), so it remains as is.
• On the right side, you didn't have to combine anything this time, but you'd simplify any constants, if present.

Combining like terms helps streamline multiple terms into fewer, simpler ones. This makes it easier for you to isolate and solve for the variable in the next steps.

Isolate Variable

The ultimate goal of solving a linear equation is to find the value of the variable. This is achieved by isolating the variable on one side of the equation. After preparing with distribution and combining like terms, the following steps are usually taken.Let's look at our equation from previous steps:

• \( 3x - 18 = -6 \)

To isolate \( x \), you need to:

• Get rid of any constant terms on the same side as \( x \). Here, you add 18 to both sides to eliminate \(-18\).
• Divide the remaining coefficient of \( x \) to solve for the variable. Dividing both sides by 3, you get \( x = 4 \).

These steps remove everything except \( x \) from one side of the equation, effectively isolating the variable. This process of addition, subtraction, multiplication, or division helps achieve the goal of solving the linear equation.