Problem 23 - Algebra 1: Concepts and Skills

Question

SOLVING EQUATIONS Use division to solve the equation. $$ 7 y=-56 $$

Step-by-Step Solution

Answer

The solution to the equation is y = -8.

Step 1: Understand the Problem

We have an equation 7y = -56. We're asked to solve this equation which means we need to isolate y, our variable of interest. We can do this by performing the inverse operations to get y alone.

Step 2: Divide both sides by 7

The current equation has 7 multiplied with y. To isolate y, we need to perform the inverse operation. That means we can remove the 7 that is being multiplied with y by dividing it out. We do this by dividing both sides of the equation by 7: \( 7y / 7 = -56 / 7 \).

Step 3: Simplify

Upon conducting the division operation we get \( y = -8 \). This means that y equals -8 is the solution to our original equation.

Key Concepts

Division in Equations

When solving equations that involve multiplication, division can be a handy tool to isolate the variable you are interested in. In our original exercise, the equation is given as \( 7y = -56 \). Here, 7 is multiplied by \( y \), and to solve for \( y \), you need to "undo" this multiplication. That's where division comes in.

By dividing both sides of the equation by 7, you're using division to counteract the multiplication, isolating \( y \) on one side of the equation.

• Original equation: 7y = -56
• Divide both sides by 7: \( \frac{7y}{7} = \frac{-56}{7} \)
• Simplified result: \( y = -8 \)

In mathematical terms, division acts as the inverse operation to multiplication. This action helps break down complex equations into simpler parts, making them easier to solve. Remember, whatever operation you do to one side of the equation, you must do to the other to keep the equation balanced.

Inverse Operations

Inverse operations are like mathematical opposites. They help us solve equations by "undoing" actions already applied to a variable. If you think about addition and subtraction, or multiplication and division, they come to mind as natural pairs of inverse operations. When an operation has been applied to a variable, performing its inverse effectively cancels out that operation.

In our exercise, the inverse operation is crucial. Because \( 7 \times y \) is part of the equation, the logical step is to divide by 7 to reverse this multiplication. This crucial step helps us isolate the variable \( y \) because dividing by 7 counteracts the multiplication:

• Start with \( 7y = -56 \)
• Perform inverse operation: Divide by 7
• Result: \( y = -8 \)

By grasping the concept of inverse operations, solving equations becomes a more straightforward process. It acts as a "key" to unlocking the variable by systematically cancelling out every operation standing in your way.

Simplifying Equations

Simplifying equations is a vital part of solving them. It involves making the equation as straightforward as possible to find the variable in question. In the exercise example \( 7y = -56 \), after deciding to divide both sides by 7, we turned our focus on simplifying the equation.

The division simplified the equation significantly. Once you've divided \(-56\) by \(7\), you are left with \( y = -8 \). This conclusion means the equation has been stripped down to an understandable format where \( y \) can easily be identified.

• First equation: 7y = -56
• Divide both sides by 7: simplifies to y = -8

Simplifying equations helps you reach a point where the solution is evident, or on easily manageable terms. It’s like clearing the clutter around the variable you need to solve. So, whenever you're solving equations, think of simplifying as a guiding light that brings clarity to the process.