Problem 8 - 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. $$-4 y=32$$

Step-by-Step Solution

Answer

The solution to the equation -4y=32 is y = -8.

Step 1: Isolate the variable

Beginning with the equation \(-4y=32\), isolate 'y' by dividing both sides of the equation by -4. The multiplication property of equality allows this operation to be performed without affecting the initial equation. It results in the equation \[y=\frac{32}{-4}\].

Step 2: Solve for y

Solving the equation \[y=\frac{32}{-4}\] gives \[y=-8\]. This is the solution to the equation.

Step 3: Verify your solution

To check the solution, substitute the found value for y into the original equation. This means the left-hand side of the equation becomes \(-4*(-8)\) which equals 32. Since the right-hand side of the equation is also 32, the solution is correct.

Key Concepts

Solving Equations

When we talk about solving equations, we're diving into the process of finding the value of the unknown variable that makes the equation true. This is like solving a puzzle, where each equation gives you a clue to reveal the missing piece, or in other words, the value of the variable.

The multiplication property of equality is a key tool here. What this property tells us is that if you multiply both sides of an equation by the same non-zero number, the two sides remain equal. This is crucial when you are manipulating equations to isolate variables and find solutions. For example, when you have an equation like \(-4y=32\), you can divide or multiply both sides to maintain equality and move towards isolating the variable. Thus enabling us to systematically reach the solution.

Isolating the Variable

Isolating the variable in an equation involves manipulating the equation in such a way that the variable you are solving for is on one side of the equation, typically by itself. Think of it as peeling away layers of an onion until you're left with the core, which in this context, is the variable by itself.

In the given exercise with the equation \(-4y = 32\), the process begins by using the multiplication property of equality — often called division in this context. You divide both sides by \-4\, the coefficient of \y\. This operation effectively 'undoes' the multiplication of \y\ by \-4\, allowing you to isolate \y\ on one side. This gives you the simpler equation \[y = \frac{32}{-4}\], which directly shows the value of \y\. This simplification is critical as it allows you to see the solution clearly and proceed to verification.

Checking Solutions

Checking solutions is a vital step in solving equations, as it ensures that the solution you found truly satisfies the original equation. It’s like reheating a dish to see if it's cooked thoroughly. When you substitute the variable back into the original equation, you confirm that the equalities hold.

For instance, after finding \y = -8\, you substitute it back into the original equation \(-4y = 32\). This becomes \(-4*(-8) = 32\), which correctly simplifies to \32 = 32\. Seeing that both sides of the equation balance confirms your solution is correct. This not only reinforces your algebraic skills but also builds confidence in your ability to solve similar problems in the future, knowing your solution checks out accurately.