Problem 74 - Prealgebra

Question

The problems below review some of the material on solving equations. Reviewing these problems will help you with the next section. Solve. $$-4 y=28$$

Step-by-Step Solution

Answer

The solution is \(y = -7\).

Step 1: Understand the Equation

The given equation is \[-4y = 28.\]In this equation, \(y\) is the variable, and it is being multiplied by \(-4\). Our task is to find the value of \(y\).

Step 2: Isolate the Variable

To isolate \(y\), we need to divide both sides of the equation by \(-4\) because \(y\) is currently being multiplied by \(-4\).\[\frac{-4y}{-4} = \frac{28}{-4}\]

Step 3: Simplify Both Sides

Simplifying the left side, the \(-4\) cancels out, leaving us with \(y\). On the right side, perform the division:\[\frac{28}{-4} = -7\]So the equation simplifies to:\[y = -7\]

Step 4: Verify the Solution

Substitute \(y = -7\) back into the original equation to check the solution:\[-4(-7) = 28\]Calculating the left side, \[28 = 28\]Since both sides are equal, \(y = -7\) is indeed the correct solution.

Key Concepts

Isolate the Variable

When solving equations, the first major step is to isolate the variable. This means getting the variable you are solving for all by itself on one side of the equation. In the original exercise, the equation is \[-4y = 28.\]The variable is \(y\) and it is currently being multiplied by \(-4\). To isolate \(y\), we want to "undo" this multiplication by performing the opposite operation, which is division.

• Identify the operation currently applied to the variable. In this case, \(-4\times y\), a multiplication.
• Perform the inverse operation: divide both sides of the equation by \(-4\) to get \(y\) by itself:

\[\frac{-4y}{-4} = \frac{28}{-4}\]This equation simplifies to \(y = -7\). Remember, whatever operation you perform on one side of an equation, you must also perform on the other side. This keeps the equation balanced, like a scale. Once \(y\) is isolated, it means we have identified the value of \(y\) that satisfies the equation.

Verify the Solution

After isolating the variable and finding its value, it's important to verify or check that this value actually satisfies the original equation. This step confirms that the solution is correct. In our example, we found \(y = -7\). To verify, substitute \(-7\) back into the original equation:\[-4(-7) = 28\]Now, calculate the left-hand side:

• Multiply: \(-4 \times -7 = 28\)

Compare this result, 28, to the right-hand side of the equation, which is already 28. Since both values on either side match exactly, this confirms that \(y = -7\) is indeed the correct solution. Verification is a crucial step to ensure accuracy when solving equations.

Simplify Equations

Simplifying an equation is a process that makes solving equations more manageable. The simplification process usually involves combining like terms or performing basic arithmetic operations. With our exercise:\[-4y = 28\],The main simplification needed was during the division:

• After dividing both sides by \(-4\), \[ -4y \div -4 = y \]This leaves us with \(y\) on the left side.
• Similarly, on the right side,\[28 \div -4 = -7\]So the equation simplifies to:\[y = -7\]

Simplifying equations makes it clearer and easier to understand the relationship between variables and constants within the equation. It helps in reducing complex problems into smaller, more manageable parts, ultimately leading to a solution.