Problem 17 - Algebra A Combined Function

Question

Solve each equation. See Examples 3 through \(5 .\) $$ \frac{2}{3} x+\frac{4}{3}=-\frac{2}{3} $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -3 \).

Step 1: Simplify the Equation

Given the equation: \( \frac{2}{3}x + \frac{4}{3} = -\frac{2}{3} \). Subtract \( \frac{4}{3} \) from both sides to isolate the term with \( x \) on one side. This gives: \( \frac{2}{3}x = -\frac{2}{3} - \frac{4}{3} \).

Step 2: Combine Like Terms

Combine the constants on the right side of the equation: \( -\frac{2}{3} - \frac{4}{3} = -\frac{6}{3} = -2 \). So the equation becomes \( \frac{2}{3}x = -2 \).

Step 3: Solve for x

To solve for \( x \), multiply both sides by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \). This gives: \( x = -2 \times \frac{3}{2} \).

Step 4: Simplify the Solution

Calculate \( x = -2 \times \frac{3}{2} = -3 \). This is the value of \( x \).

Key Concepts

Combining Like Terms

In the journey of solving linear equations, one crucial step is to combine like terms. This process involves simplifying expressions by merging terms that share the same variable and exponent. For instance, if two or more terms in an equation have the variable "x," you can add or subtract their coefficients to simplify the equation. Imagine having an equation like:

• \( 5x + 3x = 24 \)
• To combine the like terms, add the coefficients of \( x \): \( 5x + 3x = 8x \).
• Now your equation is simpler: \( 8x = 24 \).

This method of combining like terms effectively reduces the complexity of equations and helps you focus on isolating the variable next. It's a fundamental skill that not only streamlines calculations but also helps avoid errors in solving equations.

Isolating Variables

When solving equations, the ultimate aim is often to find the value of the variable. We achieve this by isolating the variable on one side of the equation. This ensures that the variable is by itself, making it easy to determine its value. Let’s think of the original equation:

• \( \frac{2}{3}x + \frac{4}{3} = -\frac{2}{3} \).
• Our goal is to focus on the term with "x," which requires getting rid of any constants on that side.
• Using operations like addition, subtraction, multiplication, or division, we slowly peel away the layers surrounding \( x \) until it stands alone.
• For example, you might subtract \( \frac{4}{3} \) from both sides to start isolating "x": \( \frac{2}{3}x = -\frac{6}{3} \).
• This process essentially 'clears the path' to find the value of \( x \).

Isolating variables not only builds a clearer path towards the solution but is an essential tool for all algebra-related problems.

Fractions in Equations

Dealing with fractions in equations might seem tricky at first, but with the right approach, it becomes quite simple. Fractions in equations just mean there's an additional step to consider, often involving finding a common denominator or using the reciprocal. Let's look at how to simplify an equation with fractions:

• Consider the equation: \( \frac{2}{3}x = -2 \).
• One effective approach is multiplying both sides of the equation by the reciprocal of the fraction coefficient.
• In this case, multiply both sides by \( \frac{3}{2} \), the reciprocal of \( \frac{2}{3} \), which cancels out the fraction on the left and isolates \( x \): \( x = -2 \times \frac{3}{2} = -3 \).
• The key here is ensuring that whatever you do to one side of the equation, you must do the same to the other side to maintain balance.

Understanding how to work with fractions in equations not only builds confidence in handling more complex operations but also simplifies reaching the solution efficiently.