Problem 9 - Prealgebra

Question

Solve each equation. Check your solution. $$-4=8 y-9 y+6$$

Step-by-Step Solution

Answer

The solution is \(y = 10\).

Step 1: Simplify the Equation

First, simplify the equation by combining like terms. The equation given is \[-4 = 8y - 9y + 6\]Combine the terms involving \(y\): \[-4 = (8y - 9y) + 6 = -1y + 6\]

Step 2: Isolate the Variable Term

To solve for \(y\), start by isolating the variable term \(-1y\). Subtract 6 from both sides:\[-4 - 6 = -1y\]Calculating on the left side gives:\[-10 = -1y\]

Step 3: Solve for y

Now, solve for \(y\) by dividing both sides by \(-1\) to get \[y = \frac{-10}{-1}\]which simplifies to:\[y = 10\]

Step 4: Check the Solution

Substitute \(y = 10\) back into the original equation to verify the solution:\[-4 = 8(10) - 9(10) + 6\]which simplifies to:\[-4 = 80 - 90 + 6\]Calculate the right-hand side:\[80 - 90 + 6 = -10 + 6 = -4\]Since both sides are equal, \(y = 10\) is the correct solution.

Key Concepts

Combining Like Terms

When solving equations, one of the first steps you should take is to simplify the equation as much as possible. This often involves combining like terms. Like terms are those that contain the same variable raised to the same power. For instance, in the equation \[-4 = 8y - 9y + 6\], the terms containing \(y\) are \(8y\) and \(-9y\). Combining these terms involves simple arithmetic:

• First, perform the operations with their coefficients: \(8y - 9y\).
• This results in \(-1y\).

Now the equation looks like this: \[-4 = -1y + 6\]. Once you've combined like terms, the equation is simpler and easier to solve.

Isolating Variables

The goal of isolating variables is to make the equation easier to solve by getting the variable on one side of the equation by itself. After simplifying the equation to \[-4 = -1y + 6\], we isolate \(-1y\) by getting rid of the constant on the same side.Here's how:

• Subtract 6 from both sides to move the constant to the left: \[-4 - 6 = -1y\].
• This simplifies to \[-10 = -1y\].

Isolating the variable helps you to see what the variable represents when the equation is balanced.

Checking Solutions

Once you find a solution, it is crucial to check it to ensure that it satisfies the original equation. For the equation \[-4 = 8y - 9y + 6\], the solution we found was \(y = 10\). To verify, substitute \(y = 10\) back into the original equation:

• Calculate: \(-4 = 8(10) - 9(10) + 6\).
• Simplify the terms: \[-4 = 80 - 90 + 6\].
• Compute the final result: \[-4 = -10 + 6\] which simplifies to \(-4 = -4\).

Both sides match, confirming that \(y = 10\) is indeed the correct solution.

Prealgebra Concepts

Solving equations often requires a foundation in prealgebra concepts. These include understanding operations such as addition, subtraction, multiplication, and division, as well as the properties of equality. When solving the equation \[-4 = 8y - 9y + 6\], consider:

• Operations: Know how to perform operations on both sides of the equation.
• Properties of Equality: When you perform an operation on one side of an equation, you must do the same to the other side to maintain balance. This is why we subtract or add constants and multiply or divide by coefficients.

Understanding these basic concepts is essential for effectively solving equations, as they provide the tools needed to manipulate and simplify complex expressions into solvable equations.