Problem 57 - Intermediate Algebra

Question

Solve each proportion. $$ \frac{9 z+6}{z^{2}+3 z}=\frac{7}{z+3} $$

Step-by-Step Solution

Answer

The solution is \(z = -3\).

Step 1: Cross Multiply

Our starting equation is \( \frac{9z+6}{z^{2}+3z} = \frac{7}{z+3} \). We will use cross-multiplication, which means multiplying the numerator of the first ratio by the denominator of the second ratio, and the denominator of the first ratio by the numerator of the second ratio. This results in: \((9z + 6)(z + 3) = 7(z^2 + 3z)\).

Step 2: Expand Both Sides

Now, we expand both sides of the equation. For the left side: \((9z + 6)(z + 3)\) expands to \(9z^2 + 27z + 6z + 18\), which simplifies to \(9z^2 + 33z + 18\). For the right side: \(7(z^2 + 3z)\) expands to \(7z^2 + 21z\). The expanded equation becomes: \(9z^2 + 33z + 18 = 7z^2 + 21z\).

Step 3: Move All Terms to One Side

To solve for \(z\), move all terms to one side so that the equation is set to zero. Subtract \(7z^2 + 21z\) from both sides: \(9z^2 + 33z + 18 - 7z^2 - 21z = 0\). This simplifies to: \(2z^2 + 12z + 18 = 0\).

Step 4: Simplify the Equation

Next, we simplify the quadratic equation \(2z^2 + 12z + 18 = 0\) by dividing every term by 2, resulting in: \(z^2 + 6z + 9 = 0\).

Step 5: Factor the Quadratic Equation

The quadratic equation \(z^2 + 6z + 9 = 0\) can be factored. Notice that it factors perfectly to \((z+3)(z+3) = 0\) or \((z+3)^2 = 0\).

Step 6: Solve for z

Since \((z+3)^2 = 0\), take the square root of both sides to solve for \(z\). Thus, \(z+3 = 0\) leads to \(z = -3\).

Key Concepts

Proportions

In mathematics, a proportion is an equation that states that two ratios are equal. In other words, a proportion compares two fractions or ratios to see if they represent the same relationship between numbers or quantities. Proportions can be very useful in solving problems that involve scaling, percentages, or conversions.
To determine whether two ratios are in proportion, you can use cross-multiplication. Cross-multiplication involves multiplying diagonally across the equal sign. For example, if you have a proportion \(\frac{a}{b} = \frac{c}{d}\), then cross-multiplying would give you \(a \cdot d = b \cdot c\). This technique can help simplify the problems and find unknown values.
When solving proportions, make sure both ratios are set up correctly, with compatible units or terms, before cross-multiplying. Once the equation is set, solve the equation just like any other algebraic problem.

Quadratic Equations

Quadratic equations are polynomial equations of degree two. They take the standard form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. These are important because they are widely applicable in various fields like physics, engineering, and economics.
There are several methods to solve quadratic equations:

• Factoring: This involves rewriting the quadratic equation in a product form \((px + q)(rx + s) = 0\) so you can set each factor to zero and solve for \(x\).
• Using the Quadratic Formula: Given \(ax^2 + bx + c = 0\), the solution can be found with the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the Square: This involves rewriting the quadratic equation in the form \((x + p)^2 = q\) and then solving for \(x\).

Understanding how to manipulate and solve these equations is crucial because they often appear in modeling real-world scenarios.

Factoring

Factoring is a key mathematical process useful for simplifying expressions and solving quadratic equations. Essentially, it involves expressing a polynomial as a product of its simpler components, or factors, which when multiplied give back the original polynomial.
Here's how to approach factoring a quadratic equation, such as \(z^2 + 6z + 9 = 0\):

• First, check if the quadratic is a perfect square. This means looking for expressions where it can be written as \((x + a)^2 = x^2 + 2ax + a^2\).
• For instance, \(z^2 + 6z + 9\) factors perfectly to \((z+3)^2 = 0\) because it matches the pattern where \((z+3)\) squared gives the original equation.
• Once factored, solve the equation by setting each factor equal to zero: \((z+3)(z+3) = 0\).
• This results in \(z+3 = 0\), solving for \(z\) gives \(z = -3\).

The simplicity of factoring makes it a preferred method when applicable. It's a powerful technique for simplifying equations and making them easier to solve.