Problem 35 - Algebra 2

Question

Solve each equation. Check your solution. $$ 27=-9(y+5)+6(y+8) $$

Step-by-Step Solution

Answer

The solution is \(y = -8\).

Step 1: Distribute and Expand

Start by distributing the coefficients \(-9\) and \(6\) through the parentheses. Multiply each coefficient by every term inside its corresponding parentheses:\[-9(y+5) = -9y - 45\]\[6(y+8) = 6y + 48\]The equation now becomes:\[27 = -9y - 45 + 6y + 48\]

Step 2: Combine Like Terms

Next, combine the like terms on the right side of the equation. Group the \(-9y\) and \(6y\) terms, as well as the constant terms \(-45\) and \(48\).\[27 = (-9y + 6y) + (-45 + 48)\]This simplifies to:\[27 = -3y + 3\]

Step 3: Isolate the Variable Term

Subtract \(3\) from both sides to isolate the term with \(y\):\[27 - 3 = -3y + 3 - 3\]This simplifies to:\[24 = -3y\]

Step 4: Solve for y

Divide both sides by \(-3\) to solve for \(y\):\[y = \frac{24}{-3}\]\[y = -8\]

Step 5: Check the Solution

Substitute \(y = -8\) back into the original equation to verify the solution:\[27 = -9(-8+5) + 6(-8+8)\]Calculate inside the parentheses first:\[27 = -9(-3) + 6(0)\]Then, multiply and sum up:\[27 = 27 + 0\]\[27 = 27\]The equality holds true, so the solution is verified.

Key Concepts

Understanding the Distributive Property

The distributive property is a helpful algebraic tool. It allows you to multiply a number by a sum or difference inside parentheses. This property is especially useful when simplifying expressions and solving equations. For example, in our equation, you see terms like -9(y+5). To distribute correctly, multiply -9 by each term inside the parentheses:

• -9 multiplied by y gives -9y
• -9 multiplied by 5 gives -45

This operation changes the expression to -9y - 45. Similarly, apply the same approach to 6(y+8) by distributing 6, resulting in 6y + 48. Always remember, using the distributive property simplifies complex expressions and sets the stage for further solving steps.

The Art of Combining Like Terms

Combining like terms is a fundamental skill in algebra. It involves merging terms that have the same variable raised to the same power. Recognizing like terms helps in reducing the complexity of equations.
In our example, after using the distributive property, the equation becomes 27 = -9y - 45 + 6y + 48. Notice the terms -9y and 6y both contain the variable y. We can combine them as follows:

• -9y plus 6y equals -3y

Similarly, look for constant terms -45 and 48. Combining these gives:

• -45 plus 48 equals 3

This simplifies the equation further to 27 = -3y + 3. Combining like terms makes equations easier to solve as it reduces the number of separate terms.

Variable Isolation and Solving

Once you've simplified the equation through distributing and combining, the next goal is variable isolation, focusing on solving for the unknown. Variable isolation means to get the variable, in this case y, by itself on one side of the equation.
In our equation, 27 = -3y + 3, start by removing constants from the right side. Subtract 3 from both sides of the equation:

• 27 minus 3 equals 24, simplifying to 24 = -3y

Now, deal with the coefficient of y, which is -3. Divide both sides by -3 to isolate y:

• 24 divided by -3 simplifies to y = -8

Variable isolation makes it clear what value the variable must take to satisfy the equation.

Solution Verification

Checking solutions ensures confidence in your answer. It's a crucial step to confirm that no errors were made in the earlier parts of the solution. To verify, substitute the solution back into the original equation.
Replace y with -8 in 27 = -9(y+5) + 6(y+8):

• Calculate inside the parentheses: -9(-8+5) and 6(-8+8)
• This becomes -9(-3) and 6(0)
• MULTIPLY: -9 multiplied by -3 equals 27 and 6 multiplied by 0 equals 0
• Add the results: 27 plus 0 equals 27

Since both sides equal, the solution y = -8 is verified as correct. Always verify to protect against simple mistakes and to ensure a solid understanding.