Problem 83 - Introductory and Intermediate Algebra for College Students 4th

Question

Exercises \(82-84\) will help you prepare for the material covered in the next section. In each exercise, solve for \(y\) and put the equation in slope- intercept form. $$y+3=-\frac{3}{2}(x-4)$$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = -\frac{3}{2}x +3\).

Step 1: Distribute the constant on the right

Distribute \(-\frac{3}{2}\) to both terms within the parenthesis on the right side of the equation, to obtain \(y+3= -\frac{3}{2}x +6\).

Step 2: Isolate \(y\)

Subtract \(3\) from both sides of the equation to isolate \(y\). This gives the final equation in slope-intercept form: \(y = -\frac{3}{2}x +3\).

Key Concepts

Solving Equations

When solving equations, we aim to find the value of the variable that makes the equation true. In this case, we are solving for \( y \). The equation is given as \( y+3=-\frac{3}{2}(x-4) \). This is a linear equation, which is often presented in different forms, such as the standard form or slope-intercept form.

The first step in solving any equation is to simplify the terms as much as possible. This usually involves distributing any constants through brackets and combining like terms. Sometimes, you may need to perform operations such as addition, subtraction, multiplication, or division on both sides of the equation, to maintain equality.

The ultimate goal is to rearrange the equation so it expresses one variable in terms of the others, often solving for \( y \) or \( x \) in algebraic problems. Once the equation is simplified, it becomes easier to isolate the desired variable.

Distributive Property

The distributive property plays a crucial role in solving equations, especially when brackets are involved. The principle can be stated as: \( a(b + c) = ab + ac \). In this problem, the distributive property helps to simplify the equation by eliminating the parentheses.

In our original problem, the term \(-\frac{3}{2}(x-4)\) requires distribution. Applying the property:

• Distribute \(-\frac{3}{2} \) to \( x \) resulting in \(-\frac{3}{2}x \).
• Distribute \(-\frac{3}{2} \) to \( -4 \) resulting in \( +6 \).

Combining these, we convert the equation from \( y+3=-\frac{3}{2}(x-4) \) to \( y+3=-\frac{3}{2}x + 6 \).

This step simplifies the equation and prepares it for isolating the variable \( y \). Using the distributive property efficiently is fundamental to managing and simplifying equations.

Isolating Variables

Isolating the variable means rearranging the equation so that the variable of interest, \( y \), stands alone on one side of the equation. It is a key part of solving linear equations and is crucial for rewriting formulas or expressions.

In the equation \( y+3=-\frac{3}{2}x + 6 \), our objective is to have \( y \) on its own. To achieve this, we need to "undo" the addition by subtracting \( 3 \) from both sides:

• Subtracting \( 3 \) from the left side leaves \( y \) alone: \( y+3-3 = y \).
• Subtracting \( 3 \) from the right side modifies the equation to \(-\frac{3}{2}x + 6 - 3 = -\frac{3}{2}x + 3 \).

After simplifying, we find that \( y = -\frac{3}{2}x + 3 \).

Thus, the equation is now in the desired slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Understanding how to isolate variables is critical for problem-solving in algebra and other areas of mathematics.