Problem 45 - Algebra 1

Question

Write the point-slope form of the equation of the line that passes through the point and has the given slope. Then rewrite the equation in slope-intercept form. $$ (-1,1), m=-\frac{1}{8} $$

Step-by-Step Solution

Answer

The point-slope form equation of the line is \(y - 1 = -\frac{1}{8}(x + 1)\) and the slope-intercept form equation of the line is \(y = -\frac{1}{8}x + \frac{7}{8}\)

Step 1: Identify the slope and the point

The problem has given the slope \(m\) as \(-\frac{1}{8}\), and the point as \((-1,1)\)

Step 2: Substitute into the point-slope form equation

Using the point-slope formula, \(y - y_1 = m(x - x_1)\), substitute \(-\frac{1}{8}\) for \(m\), \(-1\) for \(x_1\), and \(1\) for \(y_1\). This gives \(y - 1 = -\frac{1}{8}(x + 1)\)

Step 3: Simplify the equation

Solving for \(y\) we have \(y = -\frac{1}{8}x - \frac{1}{8} + 1\).

Step 4: Convert to slope-intercept form

The equation now needs to be converted to slope-intercept form, \(y = mx + b\). The equation from step 3 can be rewritten as \(y = -\frac{1}{8}x + \frac{7}{8}\)

Key Concepts

Understanding Slope-Intercept Form

The slope-intercept form is a straightforward way to write linear equations. It's expressed as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) denotes the y-intercept. This means that if you know the steepness and direction of a line (slope) and where it intersects the y-axis (y-intercept), you can graph the line or write its equation with ease. For example, with a slope of \(-\frac{1}{8}\) and a y-intercept of \(\frac{7}{8}\), the equation is plainly written as \( y = -\frac{1}{8}x + \frac{7}{8} \).

This format is particularly useful because it gives immediate visual clues about the line's behaviour on a graph. The slope tells you how sharp the incline or decline is, and the intercept shows the starting point on the y-axis.

Linear Equations: The Basis of Algebraic Graphs

Linear equations form the backbone of algebra and appear as straight lines when graphed on a coordinate plane. They can be written in various forms, such as standard form \(Ax + By = C\), point-slope form \(y - y_1 = m(x - x_1)\), and the slope-intercept form \(y = mx + b\).

In our textbook example, the equation began in point-slope form using a given point on the line \((-1,1)\) and the slope \(-\frac{1}{8}\). By applying the point-slope equation, we established a relationship between any point \((x, y)\) on the line and the given point. Simplifying that equation led us to the slope-intercept form, which is usually preferred for its directness and ease of use regarding graph plotting and interpreting the line's properties.

Algebraic Expressions and Their Role in Equations

Algebraic expressions are the combinations of numbers, variables, and arithmetic operations without an equality sign. They can simplify or complicate equations, depending on their composition and the operations involved. The art of algebra involves manipulating these expressions to isolate variables and solve equations. In the point-slope form of the equation \(y - 1 = -\frac{1}{8}(x + 1)\), simplifying the algebraic expression on the right side is necessary to reach the more approachable slope-intercept form.