Problem 60 - Algebra A Combined Function

Question

\(7(2 x+1)=18 x-19 x\)

Step-by-Step Solution

Answer

\(x = \frac{-7}{15}\)

Step 1: Distribute the 7

First, we need to distribute the 7 to both terms inside the parentheses in the expression on the left-hand side: \(7(2x + 1)\). This means multiplying both \(2x\) and \(1\) by 7. This gives us: \(14x + 7\).

Step 2: Simplify the Right Side

Next, simplify the right-hand side of the equation: \(18x - 19x\). This results in \(-x\) because \(18x - 19x = -1x\).

Step 3: Combine Like Terms

Now our equation looks like this: \(14x + 7 = -x\). We will move all terms involving \(x\) to one side and constant terms to the other.

Step 4: Move x to One Side

Add \(x\) to both sides to get rid of the negative \(x\) on the right: \(14x + x + 7 = 0\) which simplifies to \(15x + 7 = 0\).

Step 5: Isolate the x-term

Subtract 7 from both sides to isolate terms with \(x\): \(15x = -7\).

Step 6: Solve for x

Finally, divide both sides by 15 to solve for \(x\): \(x = \frac{-7}{15}\).

Key Concepts

Distributive Property

When dealing with equations that contain parentheses, the distributive property is essential for simplifying expressions. The distributive property states that multiplying a single term by two or more terms inside a set of parentheses is the same as multiplying each term individually and then adding the results. In the original exercise, we have the expression

• \( 7(2x + 1) \)

This expression requires us to multiply both \( 2x \) and \( 1 \) by \( 7 \). When applying the distributive property, you perform the following calculations:

• \( 7 \times 2x = 14x \)
• \( 7 \times 1 = 7 \)

As a result, the expression simplifies to \( 14x + 7 \). Utilizing the distributive property correctly ensures that equations are simplified accurately right from the start, setting a strong foundation for further simplification.

Simplifying Expressions

Simplifying expressions involves combining like terms and reducing an equation to its simplest form. Like terms are terms that have the same variables raised to the same power. In this exercise, we can see simplifying in action on the right-hand side of the equation:

• \( 18x - 19x \)

Both terms contain the variable \( x \), so they can be combined by subtracting the coefficients. This means:

• \( 18x - 19x = -1x \)
• Which simplifies further to \( -x \)

After distributing and simplifying both sides, our equation appears as \( 14x + 7 = -x \). The main objective here is to make the expressions as concise as possible, helping you identify how to move forward. This practice not only helps solve the equation but also enhances understanding of algebraic manipulation.

Solving for x

Solving for \( x \) in an equation means rearranging the equation so that \( x \) is isolated on one side. After distribution and simplification, our task becomes to isolate \( x \) fully. Starting with:\( 14x + 7 = -x \)We first need to get all terms involving \( x \) on one side. Add \( x \) to both sides:

• \( 14x + x + 7 = 0 \)
• Which combines to \( 15x + 7 = 0 \)

Next, isolate the \( x \)-term by moving constant terms to the opposite side. Subtract \( 7 \) from both sides:\( 15x = -7 \)Finally, divide both sides by \( 15 \) to solve for \( x \):\( x = \frac{-7}{15} \)Each step involves balancing the equation by performing identical operations on both sides. Understanding these steps is crucial in solving linear equations efficiently. This methodical approach guarantees clarity and precision when finding the value of unknown variables.