Problem 43 - Algebra 1

Question

Solve the equation. $$\frac{3}{4} z=-5 \frac{1}{2}$$

Step-by-Step Solution

Answer

The solution to the equation is \( z = -\frac{22}{3} \)

Step 1: Identify the Equation

The given equation is \( \frac{3}{4} z = -5 \frac{1}{2} \). The goal is to solve for \( z \), which means isolating \( z \) on one side of the equation.

Step 2: Convert the Mixed Number into an Improper Fraction

-5 1/2 is a mixed number, which needs to be converted into an improper fraction for proper calculation. An improper fraction is obtained by (integer * denominator) + numerator. So, -5 1/2 equals to \( -\frac{11}{2} \). Thus, the equation becomes \( \frac{3}{4} z = -\frac{11}{2} \)

Step 3: Isolating \( z \)

The current goal is to isolate \( z \) on one side of the equation. Multiply both sides of the equation by the reciprocal of \( \frac{3}{4} \), which is \( \frac{4}{3} \). The equation then becomes \( z = -\frac{11}{2} * \frac{4}{3} \)

Step 4: Simplify the Equation for the Solution

Multiplying the fractions on the right side gives the solution to the equation. \( z = -\frac{22}{3} \)

Key Concepts

Isolating Variables

When solving algebraic equations, one of the fundamental steps is to isolate the variable you are solving for. This means manipulating the equation so that the variable is alone on one side of the equality sign. The other side of the equation should contain the numbers or expressions that state the value of the isolated variable.

For an equation like \( \frac{3}{4} z = -5\frac{1}{2} \), you want to get \( z \) by itself on one side. To achieve this, you can perform a variety of operations, such as adding, subtracting, multiplying, or dividing both sides of the equation by the same number, which does not change the equation's balance. The key is performing the inverse operation of what is currently being applied to the variable.

In the example, \( z \) is being multiplied by \( \frac{3}{4} \). To undo this multiplication, you multiply both sides by the reciprocal of \( \frac{3}{4} \), which is \( \frac{4}{3} \). This 'cancels out' the \( \frac{3}{4} \) on the left, leaving \( z \) isolated.

Converting Mixed Numbers to Improper Fractions

Mixed numbers are a combination of a whole number and a fraction. Converting mixed numbers to improper fractions is an essential skill when solving equations with fractions. An improper fraction has a numerator (the top number) that is larger than or equal to its denominator (the bottom number).

The conversion process is straightforward but crucial for carrying out operations such as multiplication or division with fractions. To convert a mixed number to an improper fraction, you multiply the whole number by the denominator of the fractional part, then add the numerator of the fractional part. The resulting sum becomes the new numerator, and you keep the original denominator.

For instance, in our exercise \(-5\frac{1}{2}\), you would multiply 5 (the whole number) by 2 (the denominator) to get 10, and then add 1 (the numerator) to get 11. Thus, \(-5\frac{1}{2}\) is equivalent to \(-\frac{11}{2}\). This conversion is vital to solving the equation in a more uniform manner.

Multiplying Fractions

Multiplication of fractions is a key operation in algebra. The rule for multiplying fractions is simple: you multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator.

Unlike addition and subtraction of fractions, there's no need to have a common denominator. To multiply fractions, you follow the formula \( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \). This process can also include simplifying the resulting fraction by dividing both the numerator and denominator by their greatest common divisor (GCD).

In our example, the multiplication step required to isolate \( z \) is \(-\frac{11}{2} \times \frac{4}{3}\). The numerators 11 and 4 are multiplied to get 44, and the denominators 2 and 3 are multiplied to get 6, which would then usually be simplified. Since 44 and 6 share a common factor of 2, the fraction would be simplified to \(-\frac{22}{3}\), which is the solution for \( z \).