Problem 49 - Algebra 1: Concepts and Skills

Question

Which equation is in point-slope form? A. \(y=5 x-9\) B. \(y+4=3(-2 x+2)\) C. \(x=8(y-1)\) D. \(y+4=3\left(x-\frac{3}{2}\right)\)

Step-by-Step Solution

Answer

In conclusion, none of these equations are in the point-slope form. It seems like there might be some typographical error in the options provided for this question

Step 1: Inspect Option A

Option A is \(y=5x-9\). This equation is in slope intercept form where slope is 5 and y-intercept is -9. Therefore, this is not in point-slope form.

Step 2: Inspect Option B

Option B is \(y+4=3(-2x+2)\). We can simplify it and get \(y=-6x+6-4\) which in essence is \(y=-6x+2\). This equation is also not in point-slope form

Step 3: Inspect Option C

Option C is \(x=8(y-1)\). After rearranging the equation, we get \(y=\frac{1}{8}x+1\). This equation is again not in point-slope form as it is already in the form \(y=mx+b\).

Step 4: Inspect Option D

Option D is \(y+4=3\left(x-\frac{3}{2}\right)\). This equation looks similar to point-slope form. After simplifying, it can be written as \(y+4=3x-\frac{9}{2}\) or \(y=3x-\frac{9}{2}-4\) which in essence is \(y=3x-\frac{17}{2}\). So, this equation is not in point-slope format as well

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a simple way to describe straight lines. This form is given by the equation \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept, the point where the line crosses the y-axis. This format is incredibly useful for quickly identifying these two key characteristics of a line.
Before diving deeper, let's briefly explain the components:

• **Slope (\(m\))**: It shows how steep the line is. A positive slope means the line rises from left to right, while a negative slope means it falls.
• **Y-intercept (\(b\))**: It is the starting point of the line on the y-axis.

When given a linear equation, converting it into the slope-intercept form allows for easy graphing and helps in predicting future values. For example, Option A from the exercise, \(y = 5x - 9\), shows a slope of 5, meaning a rapid incline, and a y-intercept of -9, indicating that the line crosses the y-axis below the origin.

Linear Equations

Linear equations represent lines in a two-dimensional space and are characterized by a constant rate of change, the slope. These equations can be written in different forms, such as point-slope, slope-intercept, and standard forms. The primary feature of linear equations is that they have no exponents higher than one.
Linear equations usually aim to describe the relationship between two variables, most commonly \(x\) and \(y\). A crucial aspect is understanding that the solutions of these equations are infinite points on the line described by the equation. Therefore, every combination of \(x\) and \(y\) that satisfies the equation will plot a point on a graph that contributes to forming the line.
This characteristic makes linear equations essential in various fields such as economics for demand curves, physics for velocity, and everyday decision-making scenarios like budgeting.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to gain insights or make them easier to solve. This process includes combining like terms, expanding expressions, and isolating variables.
Let's take a look at how algebraic manipulation applies to the options in the original exercise:

• In Option B, \(y+4=3(-2x+2)\), the manipulation simplified this to \(y=-6x+2\).
• Similarly, Option C's equation \(x=8(y-1)\) was rearranged to match the slope-intercept form \(y=\frac{1}{8}x+1\).
  Through these steps, we clarify and transform equations into more standard forms for analysis.

These techniques are pivotal in solving algebraic equations and understanding relationships in mathematics, ensuring that even complex problems can be broken down into manageable parts.