Problem 63 - Algebra A Combined Function

Question

Solve each equation for y. See Section 2.5. $$ y-1=-6(x-(-2)) $$

Step-by-Step Solution

Answer

The solution for \( y \) is \( y = -6x - 11 \).

Step 1: Rewrite the Equation

First, let's rewrite the given equation. The original equation is:\[ y - 1 = -6(x - (-2)) \]We can simplify the expression \( x - (-2) \) to \( x + 2 \). So, the equation becomes:\[ y - 1 = -6(x + 2) \]

Step 2: Expand the Right Side

Next, expand the right-hand side of the equation by distributing the \(-6\):\[ y - 1 = -6x - 12 \]Now the linear equation is expressed with \(-6x\) and \(-12\) on the right side.

Step 3: Isolate y

Now, to solve for \( y \), isolate \( y \) on the left-hand side. Add 1 to both sides of the equation:\[ y - 1 + 1 = -6x - 12 + 1 \]This simplifies to:\[ y = -6x - 11 \]

Step 4: Final Check

Finally, let's ensure the expression is correct. The value of \( y \) is now isolated and the equation is simplified. Hence, the solution meets the requirements of isolating \( y \).

Key Concepts

Solving Equations

Solving equations is a fundamental process in algebra, where we find the value of one or more unknown variables. To solve an equation means to determine the value of the unknown that makes the equation true. When solving equations, it's important to perform operations that systematically isolate and find the values of these unknowns.

Here are some key steps typically involved in solving equations:

• Simplify both sides of the equation if necessary. This might include distributing terms, combining like terms, or simplifying complex fractions.
• Look for opportunities to apply inverse operations, such as adding or subtracting, to both sides of the equation. This will help in moving terms and simplifying the process of finding the unknown.
• Continue to isolate the variable on one side of the equation by performing equivalent operations on both sides, ensuring the equation stays balanced.
• Check your solution by plugging back the value into the original equation to verify that both sides are equal.

Solving equations is not only an essential skill for algebra but also forms the foundation for more advanced mathematical concepts.

Isolating Variables

Isolating variables is a crucial technique in algebra, used to solve equations by getting the variable alone on one side of the equation. This allows us to see clearly what the variable equals.

To isolate a variable:

• Identify the variable you need to isolate. In linear equations, this often involves isolating either "x" or "y".
• Perform algebraic operations to both sides of the equation. For example, if an equation is in the form of \( y - 1 = -6x - 12 \), you can isolate \( y \) by adding 1 to both sides.
• Simplify the equation as you progress. This involves combining like terms and simplifying any arithmetic.

Isolating variables effectively is critical because it leads to defining the relationship between dependent and independent variables, especially in linear equations.

Linear Equations

Linear equations are equations of the first degree, meaning they involve variables raised only to the power of one. These types of equations form a straight line when graphed on a coordinate plane and are fundamental in algebra and calculus.

Characteristics of linear equations include:

• They take the general form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.
• Their graphs are straight lines.
• They have constant slopes that determine the steepness of the line.

Histograms, distances, and social predictions often utilize linear equations. Recognizing linear equations can help simplify complex situations because they only have one solution or, in the case of intersecting lines, a simple relationship between two variables.