Problem 62 - Algebra A Combined Function

Question

Solve each equation for y. See Section 2.5. $$ y-7=-9(x-6) $$

Step-by-Step Solution

Answer

The solution is \( y = -9x + 61 \).

Step 1: Distribute the Right Side

Distribute the -9 on the right side of the equation to both terms inside the parentheses. This means multiplying -9 by each of the terms inside the parentheses: \( -9(x - 6) = -9x + 54 \). So the equation becomes \( y - 7 = -9x + 54 \).

Step 2: Isolate y

To solve for \( y \), you need to get \( y \) by itself on one side of the equation. Add 7 to both sides of the equation to cancel out -7 next to \( y \):\( y - 7 + 7 = -9x + 54 + 7 \).This simplifies to \( y = -9x + 61 \).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions by multiplying a single term with each term inside a set of parentheses. For the original problem, we used the distributive property to eliminate the parentheses on the right side of the equation.Consider the equation:

• \( y - 7 = -9(x - 6) \)

To use the distributive property, multiply the \(-9\) by each term inside the parentheses:

• \(-9(x) + (-9)(-6)\)
• This results in \(-9x + 54\)

After distributing, the equation transforms into:

• \( y - 7 = -9x + 54 \)

Remember that distributing negative numbers requires attention to the signs during multiplication. This transformation is crucial for simplifying equations and solving for variables.

Isolating Variables

In algebra, isolating variables is an essential step that involves rearranging an equation to get a particular variable on one side. When solving equations for a specific variable, such as \( y \), isolating it allows you to express it in terms of other known elements of the equation.In our example:

• We started with \( y - 7 = -9x + 54 \)

To isolate \( y \) on the left side, perform operations that "undo" any subtraction, addition, multiplication, or division impacting \( y \):

• Add 7 to both sides to cancel out the \(-7\) next to \( y \)
• This operation results in the simplified equation: \( y = -9x + 61 \)

The key is to systematically undo operations until the desired variable stands alone. Performing the same operation on both sides of the equation maintains equality, ensuring the variable isolation process is accurate.

Solving for y

"Solving for \( y \)" implies adjusting an equation until \( y \) is expressed on its own in terms of other variables or constants. This allows us to see how \( y \) depends on these other factors, making it a central skill in algebra.Looking at the equation we've been solving:

• Begin with \( y - 7 = -9x + 54 \)
• Our goal is to express \( y \) on its own on one side of the equation.

To do this, add 7 to each side:

• This operation results in \( y = -9x + 61 \)

Now \( y \) is solved for, representing \( y \) as a function of \( x \). Understanding the direct relationship between \( y \) and other variables assists in modeling and interpreting real-world problems. Thus, solving for \( y \) is not just a procedural step; it provides insight into the behavior and influence of variables within an equation.