Problem 28 - Algebra 1

Question

Write an equation in point-slope form of the line that passes through the given points. $$ (-3,-7),(-4,-8) $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form that passes through the points (-3,-7) and (-4,-8) is \( y = x - 4 \)

Step 1: Calculate the Slope

The slope \( m \) of a line through two points \( (x_2, y_2) \) and \( (x_1, y_1) \) is given by the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points \( (-3, -7) \) as \( (x_2, y_2) \) and \( (-4, -8) \) as \( (x_1, y_1) \), we have \[ m = \frac{-7 - (-8)}{-3 - (-4)} = 1 \]

Step 2: Write the Equation in Point-Slope Form

Now that we have the slope, we can write the equation. You can use either of the points, and in this case, we'll use \( (-3, -7) \) as \( (x_1, y_1) \). Substituting these values into the point-slope form gives us \[ y - (-7) = 1(x - (-3)) \]

Step 3: Simplify the Equation

Simplify the equation to its final form \[ y + 7 = x + 3 \] Further simplifying by subtracting 7 from both sides gives \[ y = x - 4 \]

Key Concepts

Calculating Slope

Understanding how to calculate the slope of a line is crucial for anyone studying algebra. The slope is a measure of how steep a line is, and it's represented by the letter 'm'. To calculate the slope, pick two distinct points on a line, say \( (x_1, y_1) \) and \( (x_2, y_2) \). Apply the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

For instance, considering the points \( (-3, -7) \) and \( (-4, -8) \), insert these coordinates into the formula as follows: \[ m = \frac{-7 - (-8)}{-3 - (-4)} \] Hence, after performing the subtraction in the numerator and the denominator, we find that the slope is 1. This result indicates that for each unit the line moves horizontally, it moves up or down by the same amount vertically, thus having an equal rise to run ratio.
Calculating slope is essential as it represents the line's direction and steepness, making it a foundational step in graphing linear equations.

Writing Equations of a Line

After calculating the slope, it's important to learn how to express the equation of a line. Writing the equation begins with understanding the point-slope form, which is structured as: \[ y - y_1 = m(x - x_1) \]

This equation signifies that for any point \( (x, y) \) on the line, the difference in the y-values between that point and a known point \( (x_1, y_1) \) on the line is equal to the slope 'm' times the difference in their x-values. To demonstrate, if we have the slope \( m = 1 \) and a point \( (-3, -7) \) from the original exercise, the equation in point-slope form becomes: \[ y - (-7) = 1(x - (-3)) \]
By rearranging and simplifying, we have the linear equation relating x and y.

Choosing a Point to Use

It doesn't matter which point you pick if you have two points that the line passes through; both will yield the same line's equation.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve for variables or to present the expression in a more useful format. This skill is particularly important when simplifying equations in algebra. In regards to our example, the point-slope form we derived needs to be simplified for clarity. We work with the equation \[ y - (-7) = 1(x - (-3)) \]

Starting with removing the parentheses gives us: \[ y + 7 = x + 3 \] Then, to isolate the variable y, we perform the same operation on both sides of the equation, in this case, subtracting 7. This leads to the simplified form: \[ y = x - 4 \] The equation now clearly shows how y is calculated for any x value, which is the fundamental goal of algebraic manipulation—making equations easier to use and understand.