Problem 66 - Algebra for College Students

Question

Solve each equation by an appropriate method. $$\frac{z}{z+3}=\frac{3 z}{5 z-1}$$

Step-by-Step Solution

Answer

The solution is z = 0 or z = 5.

Step 1: Cross Multiply

To solve the equation \[\frac{z}{z+3}=\frac{3z}{5z-1},\] start by cross-multiplying: \[(z) (5z - 1) = (3z) (z + 3).\]

Step 2: Distribute the Terms

Distribute the terms on both sides of the equation: \[(z \times 5z - z \times 1) = (3z \times z + 3z \times 3).\] This simplifies to \[5z^2 - z = 3z^2 + 9z.\]

Step 3: Simplify and Collect Like Terms

Collect all terms on one side of the equation to set it equal to zero: \[(5z^2 - z) - (3z^2 + 9z) = 0.\] Simplify this to get \[2z^2 - 10z = 0.\]

Step 4: Factor the Quadratic Equation

Factor out the common term, which is z: \[z(2z - 10) = 0.\]

Step 5: Solve for z

Set each factor equal to zero and solve for z: \[z = 0\] and \[2z - 10 = 0.\] For the second equation, solve \[2z = 10.\] Divide both sides by 2: \[z = 5.\]

Key Concepts

Cross Multiplication

Cross multiplication is a powerful method to solve rational equations. It involves multiplying the numerator of one fraction by the denominator of the other fraction and doing the same in reverse. This turns the equation into a simpler one with no fractions. In our example, we had \(\frac{z}{z+3} = \frac{3z}{5z-1} \). By cross-multiplying, we get \( z \times (5z - 1) = 3z \times (z + 3)\). Now, you've turned a fraction equation into \ 5z^2 - z =3z^2 + 9z.\. Let's move on to the next step.

Distributive Property

The distributive property allows you to multiply a single term outside the parentheses by each term inside the parentheses. It states \ a(b + c) = ab + ac \. In the context of our equation, we distribute the terms on each side: \( z \times 5z - z \times 1 = 5z^2 - z \) and \ 3z \times z + 3z \times 3 = 3z^2 + 9z \. This simplifies our equation to \ 5z^2 - z = 3z^2 + 9z \. Now, our goal is to collect like terms so we can solve for \ z\.

Factoring Quadratic Equations

Quadratic equations often require factoring to find their solutions. After distributing and simplifying, we get: \[ 5z^2 - z - 3z^2 - 9z = 0 \] which simplifies to \[ 2z^2 - 10z = 0 \]. To factor this equation, we look for common factors in all terms. Clearly, \ z \ is common in both terms, so we factor out \ z \ to get: \ z(2z - 10) = 0 \.

Simplifying Equations

Simplifying equations means breaking them down to their simplest form. After factoring, we have \[ z(2z - 10) = 0 \]. To solve for \ z \, we set each factor equal to zero: \ z = 0 \ and \ 2z - 10 = 0 \. Solving \ 2z - 10 = 0 \, we get: \ 2z = 10 \, which simplifies to \ z = 5 \. Therefore, our solutions are \ z = 0 \ and \ z = 5 \.