Problem 25 - Algebra A Combined Function

Question

Solve each equation. Don't forget to first simplify each side of the equation, if possible. Check each solution. See Examples 5 through 7 . $$ 8 y+2-6 y=3+y-10 $$

Step-by-Step Solution

Answer

The solution to the equation is \(y = -9\).

Step 1: Simplify Each Side

First, we need to simplify each side of the equation, if possible. Start by combining like terms on both sides. For the left side, combine the terms with \(y\): \(8y - 6y = 2y\). So the left side becomes \(2y + 2\). The right side is already simplified as \(3 + y - 10\).

Step 2: Simplify Further

Now further simplify the expression on the right side of the equation. Combine the constant terms: \(3 - 10 = -7\). Therefore, the right side becomes \(y - 7\). The equation is now \(2y + 2 = y - 7\).

Step 3: Isolate Variable "y"

To solve for \(y\), first get all terms involving \(y\) on one side of the equation and constants on the other. Subtract \(y\) from both sides: \(2y - y + 2 = -7\), leading to \(y + 2 = -7\).

Step 4: Solve for "y"

Isolate \(y\) by subtracting \(2\) from both sides: \(y = -7 - 2\). This results in \(y = -9\).

Step 5: Check the Solution

To verify the solution, substitute \(y = -9\) back into the original equation. Check: \(8(-9) + 2 - 6(-9) = 3 + (-9) - 10\). Simplifying, \(-72 + 2 + 54 = -9 - 10\) which calculates as \(-16 = -16\). The solution satisfies the original equation.

Key Concepts

Combining Like Terms

When solving linear equations, one of the first steps is to simplify each side of the equation by combining like terms. Like terms are terms that have the same variables raised to the same power. In the given equation, both sides need to be simplified. Let's break it down easily:

• Identify Like Terms: On the left side of the equation, the terms are 8y and -6y. These are like terms because they both involve the variable y.
• Combine Them: Add or subtract the coefficients (the numbers in front of the variables) to combine them. Here, you subtract 6 from 8, resulting in 2y.
• Other Terms: With 2 left on the left side, the expression becomes 2y + 2.

On the right side, initially, we had constant numbers 3 and -10. By combining these, we get -7, making the right side y - 7. Breaking down both sides into like terms helps simplify the equation greatly, making it easier to solve.

Isolating Variables

Once we've combined like terms and simplified the equation, the next goal is to isolate the variable, in this case, y. Isolating the variable means getting y by itself on one side of the equation. Here's how to do it step-by-step:

• Move y to One Side: Subtract y from both sides of the equation, turning 2y + 2 = y - 7 into y + 2 = -7. When we moved y, we ensured all terms involving y were on one side.
• Focus on y Alone: To further isolate y, subtract 2 from both sides, resulting in y = -9. This step ensures that y is completely by itself.

These steps are crucial as they eliminate additional terms from the variable, allowing us to express y in terms of clear numeric values.

Checking Solutions

After finding an answer, such as y = -9, it's important to verify if this solution truly works in the original equation. Checking the solution helps confirm our work:

• Substitute y Back: Take y = -9 and plug it back into the original equation: 8(-9) + 2 - 6(-9) = 3 - 9 - 10.
• Solve Each Side: Solve both sides separately. Start with the left, which becomes -72 + 2 + 54 = -16. Now, for the right side, simplify it to -9 - 10 = -16.
• Compare: Check if both sides equal the same number. Here, -16 equals -16, proving the solution is correct.

By checking your solution, you ensure that no mistakes were made during solving. This step is crucial for precise and accurate problem-solving.