Problem 83 - Introductory Algebra for College Students

Question

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 Value Across Parentheses

Firstly, distribute -3/2 across (x-4), which yields \(-\frac{3}{2}x + 6\). So the equation becomes \(y + 3 = -\frac{3}{2}x + 6\)

Step 2: Isolate y

Subtract 3 from both sides of the equation to isolate \(y\). After doing this, the equation will be \(y = -\frac{3}{2}x + 3\)

Step 3: Write in Slope-Intercept Form

The equation is now in slope-intercept form: \(y = mx + b\). In this case, \(m = -\frac{3}{2}\) which is the slope, and \(b = 3\) which is the y-intercept

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and represent equations of the first degree, meaning they contain no exponents greater than one. These equations graph as straight lines on the Cartesian plane. Understanding linear equations is crucial as they are foundational for more advanced mathematics.
Linear equations typically have the form:

• Standard Form: \( Ax + By = C \)
• Slope-Intercept Form: \( y = mx + b \)

Where \( m \) is the slope and \( b \) is the y-intercept. The slope indicates the steepness of a line, whereas the y-intercept is the point where the line crosses the y-axis.
In our example, we begin with solving an equation for \( y \) to express it in the slope-intercept form, a common requirement for graphing linear equations.

Solving for Y

Solving for \( y \) in an equation involves rearranging the equation to isolate \( y \) on one side. This process often involves reversing mathematical operations that have been performed on \( y \).
In our given exercise, we begin with the equation \( y + 3 = -\frac{3}{2}(x - 4) \). The first step is to distribute \(-\frac{3}{2}\) across \((x - 4)\):

• Distributing results in: \( y + 3 = -\frac{3}{2}x + 6 \)

Next, isolate \( y \) by subtracting 3 from both sides of the equation:

• This yields: \( y = -\frac{3}{2}x + 3 \)

By completing these steps, the equation is efficiently transformed into a form that's easy to graph and understand in terms of its slope and intercept.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols such as addition or multiplication. Key operations when dealing with algebraic expressions in the context of linear equations include distribution and simplification.
In our exercise, the expression \(-\frac{3}{2}(x - 4)\) demonstrates distribution, an essential algebraic technique. Here, \(-\frac{3}{2}\) is multiplied by each term inside the parentheses. This operation transforms the expression into a linear form suitable for equation solving.

• Distribution: \(-\frac{3}{2}x + 6\)

Understanding how to manipulate algebraic expressions allows us to solve equations and rearrange them into formats, such as the slope-intercept form, that provide clear mathematical insights and graphical representations. Algebraic expressions are like building blocks, and learning to handle them is crucial for mastering algebra.