Problem 64 - 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

The given equation is: \[ y - 7 = -9(x - 6) \] First, distribute \(-9\) to both terms inside the parentheses on the right side of the equation:\[ y - 7 = -9x + 54 \]

Step 2: Isolate the y-term

To solve for \(y\), we need to isolate it on one side of the equation. Start by adding \(7\) to both sides:\[ y - 7 + 7 = -9x + 54 + 7 \]This simplifies to:\[ y = -9x + 61 \]

Key Concepts

Distributive Property

The distributive property is an essential tool in solving equations in algebra. It allows us to expand expressions and make them simpler to solve. In this exercise, the distributive property is applied to the term \(-9(x - 6)\). Here's how it works in general:- **Distribute**: Multiply the factor outside the parentheses by each term inside the parentheses.- **Example**: For \(a(b + c)\), distribute as \(ab + ac\).In the given equation, the distribution step involves applying \(-9\) to both \(x\) and \(-6\): - Multiply \(-9\) by \(x\), giving \(-9x\). - Then, multiply \(-9\) by \(-6\), resulting in \(54\). So, \(-9(x - 6)\) turns into \(-9x + 54\). This step simplifies the equation, setting it up for easier solving.

Isolating Variables

Isolating the variable is crucial when you are solving an equation, as it allows you to find the value of the unknown. Here, the goal is to solve for \(y\). To isolate \(y\), follow these steps:- **Identify**: Notice which terms are not related to \(y\) and need to move to the opposite side of the equation.- **Use Inverse Operations**: To move a term to the other side, use the inverse operation.In the transformed equation from the distribution step, \(y - 7 = -9x + 54\):- **Add 7**: Move \(-7\) to the other side by adding 7 to both sides of the equation. This operation cancels \(-7\) on the left side.Doing this will give us: \[ y = -9x + 61 \] With \(y\) isolated, you now have an expression that directly relates \(y\) to \(x\). This expression makes it much easier to understand the relationship between the variables.

Algebraic Expressions

Understanding algebraic expressions is vital for grasping how equations work. An algebraic expression combines numbers, variables, and operations. When solving equations, transforming and rearranging these expressions is key.- **Terms**: Parts of an expression separated by addition or subtraction. For instance, \(-9x, 61\) are terms.- **Coefficients**: Numbers multiplying the variables, like \(-9\) in \(-9x\).In our example, after applying the distributive property and isolating \(y\), you achieved the expression:\(y = -9x + 61\)This shows that \(y\) is expressed in terms of \(x\), where \(-9x\) indicates \(x's\) contribution to \(y\)'s value, shifted up by 61 units. Understanding how expressions like these represent relationships helps in visualizing solutions, predicting outcomes, and representing real-world situations. Therefore, mastering how to manipulate and interpret algebraic expressions is fundamental in algebra.