Problem 43 - Elementary and Intermediate Algebra

Question

Solve each proportion. $$ \frac{8 x}{3}=\frac{11 x+9}{4} $$

Step-by-Step Solution

Answer

The solution to the proportion is \( x = -27 \).

Step 1: Cross-Multiply

To solve the proportion \( \frac{8x}{3} = \frac{11x + 9}{4} \), begin by cross-multiplying to eliminate the fractions. This means we multiply the numerator of one fraction by the denominator of the other:\[ 8x imes 4 = 3 imes (11x + 9) \]This simplifies to:\[ 32x = 33x + 27 \]

Step 2: Rearrange the Equation

The next step is to move all terms involving \( x \) to one side of the equation and the constant term to the other. Subtract \( 33x \) from both sides:\[ 32x - 33x = 27 \]This simplifies to:\[ -x = 27 \]

Step 3: Solve for x

To solve for \( x \), divide both sides by -1:\[ x = -27 \]

Key Concepts

Cross-Multiplication

Cross-multiplication is a technique used to simplify equations involving fractions, especially when dealing with proportions. This method is particularly handy because it lets us eliminate the fractions, which makes solving the equation much easier. In our example, we have the equation \( \frac{8x}{3} = \frac{11x + 9}{4} \). To cross-multiply, we take the numerator of the first fraction, \( 8x \), and multiply it by the denominator of the second fraction, which is 4. Meanwhile, we multiply the numerator of the second fraction, \( 11x + 9 \), by the denominator of the first fraction, which is 3. This gives us:

• \( 8x \times 4 = 32x \)
• \( 3 \times (11x + 9) = 33x + 27 \)

Both products are then set equal to each other. Thus, we get the simplified equation: \( 32x = 33x + 27 \). Cross-multiplication helps by transforming a fraction equation into a linear form, which is simpler to work with.

Solving Proportions

Solving proportions involves finding the value of a variable that makes two ratios equivalent. In algebra, a proportion is an equation that asserts that two ratios are equal.To solve a proportion like \( \frac{8x}{3} = \frac{11x + 9}{4} \), we leverage cross-multiplication to clear the fractions. Once we use cross-multiplication, we rearrange the resulting equation. Here are the steps:

• We first cross-multiply to obtain the equation: \( 32x = 33x + 27 \).
• Next, we rearrange this equation to bring all the terms involving \( x \) on one side and constant terms on the other.

In our case, we subtract \( 33x \) from both sides, yielding:- \( 32x - 33x = 27 \)This simplifies down to \( -x = 27 \). By solving this step, we make sure that the variable is isolated and we can find its value effectively.

Linear Equations

Linear equations are mathematical expressions that represent a straight line when graphed. These equations hold the standard form \( ax + b = c \), where "\( a\)", "\( b\)", and "\( c\)" are constants, and "\( x\)" is the variable being solved.The ultimate goal is to isolate \( x \) on one side of the equation. In the example \( 32x = 33x + 27 \), once simplified to \( -x = 27 \), we use basic algebraic operations to isolate \( x \). Here’s how:

• We divide both sides by -1, resulting in \( x = -27 \).

By reaching this solution, we've shown that the variables along with their coefficients when manipulated correctly, can provide us with the exact value of \( x \) that solves the equation. Understanding linear equations is foundational as it applies to various real-world scenarios and advanced mathematics.