Problem 17 - Introductory Algebra for College Students

Question

Solve each equation in using the multiplication property of equality. Be sure to check your proposed $$\frac{3}{4} y=15$$

Step-by-Step Solution

Answer

The solution to the given equation \( \frac{3}{4} y=15 \) is \( y = 20 \).

Step 1: Understand the problem

In the given equation \( \frac{3}{4} y=15 \), the variable 'y' is being multiplied by \( \frac{3}{4} \). Using the multiplication property of equality, we can eliminate the fraction from the equation by multiplying both sides of the equation by the reciprocal of \( \frac{3}{4} \) i.e, \( \frac{4}{3} \).

Step 2: Apply the multiplication property of equality

Multiply both sides of the equation by \( \frac{4}{3} \): \( \frac{4}{3} \times \frac{3}{4} y = \frac{4}{3} \times 15 \). Simplifying this yields, \( y = 20 \). So the solution to the equation is \( y = 20 \).

Step 3: Check the solution

Substitute the value of y in the original equation to verify if both sides of the equation are equal. Plugging in \( y = 20 \) into the original equation results in \( \frac{3}{4} \times 20 = 15 \) which simplifies to \( 15 = 15 \). Since the left side of the equation equals the right side, the solution is verified.

Key Concepts

Solving Algebraic Equations

When it comes to solving algebraic equations, understanding how to manipulate the equation is crucial. The goal is to isolate the variable on one side of the equation to find its value. Start by simplifying the equation, combining like terms, and using inverse operations to cancel out any addition, subtraction, multiplication, or division that is being applied to the variable. Always perform the same operation on both sides of the equation to maintain equality—this is known as the multiplication property of equality.

For instance, in the equation \( \frac{3}{4} y = 15 \) we are dealing with a fractional coefficient. The multiplication property of equality tells us that multiplying both sides of the equation by the same non-zero number will not change the solution set of the equation. Thus, by multiplying both sides by the reciprocal of \( \frac{3}{4} \), we eliminate the fraction and make the equation easier to solve. This brings us a step closer to finding the value of 'y'.

Fractional Coefficients

Fractional coefficients often intimidate students, but they follow the same rules as integer coefficients. When a variable has a fractional coefficient, you can use the reciprocal of the fraction to cancel it out. The key to working with fractions is to remember that multiplying by the reciprocal yields a product of 1, which effectively removes the fraction from the variable.

Let's take \( \frac{3}{4} y = 15 \) as an example. The reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \) since \( \frac{3}{4} \times \frac{4}{3} = 1 \). By multiplying both sides by \( \frac{4}{3} \) (the reciprocal of our fractional coefficient), we've cleverly converted the fractional equation into a simpler one: \( y = 20 \). It's like magic, but it's actually just mathematics!

Equation Verification

Equation verification is a critical step to ensure that the solution you've found is correct. To verify a solution, substitute the variable in the original equation with the value obtained. If both sides of the equation equal the same number after substitution, then the solution is verified.

For the equation \( \frac{3}{4} y = 15 \), after solving it we found that \( y = 20 \). To check our work, we plug 20 back into the original equation: \( \frac{3}{4} \times 20 \). If we execute the multiplication, we get \( 15 = 15 \), proving that our solution is indeed correct. Verification not only confirms the accuracy of the solution but also reinforces your understanding of the problem-solving process. Never skip this critical step, as it is your assurance that you've solved the equation properly.