Problem 15 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation using the multiplication property of equality. Be sure to check your proposed solutions. $$17 y=0$$

Step-by-Step Solution

Answer

The solution to the equation \(17y = 0\) is \(y = 0\).

Step 1: Divide both sides of the equation

In order to isolate \(y\) on one side of the equation, divide both sides of the \(17y = 0\) by 17. The equation becomes \(y = 0 / 17\).

Step 2: Solve for y

Solving for \(y\) by dividing 0 by any number will always be 0. So, \(y = 0\).

Step 3: Check the proposed solution

To check if the proposed solution is correct, substitute \(y = 0\) back into the original equation \(17y = 0\). If both sides of the equation are equal, then the proposed solution is correct. Thus \(17*0 = 0\), hence the solution \(y = 0\) is correct.

Key Concepts

Solving Algebraic Equations

Understanding how to solve algebraic equations is a fundamental skill in mathematics. An equation is a statement that two expressions are equal, and to solve an equation means to find the value of the unknown variable that makes the statement true. One common method is using the multiplication property of equality. This property states that if you multiply both sides of an equation by the same nonzero number, the two sides remain equal.

Take the equation from our original problem:
\[17y = 0\].

To isolate the variable \(y\), we divide both sides by 17, since division is the inverse operation of multiplication. This gives us:

Properties of Zero

The properties of zero play a crucial role in understanding algebra. To start with, any number multiplied by zero is zero, which is why when we divide \(0\) by any nonzero number, the result is still \(0\). This leads to our solution where \(y\) is indeed zero in the given equation \(17y = 0\). This unique property simplifies many algebraic operations and helps in quickly finding solutions to equations involving multiplication or division by zero.

Furthermore, zero is the only number that, when added to or subtracted from another number, leaves the original number unchanged, a useful tip for rearranging and solving equations.

Checking Solutions in Algebra

After calculating a proposed solution for an algebraic equation, it is essential to verify its accuracy. This is done by performing a 'check'—replacing the variable in the original equation with the proposed solution. If both sides of the equation maintain equality after this substitution, the solution is verified.

In our example, we substitute \(y = 0\) back into the original equation to get \(17 \times 0 = 0\). Since both sides equal zero, we can confidently say our solution is correct. This process not only validates our answer but also reinforces our understanding of algebraic principles and helps avoid errors in solving. Implementing this step is an excellent habit for all students to adopt.