Problem 13 - Algebra 1

Question

Write an equation in slope-intercept form of the line that passes through the points. $$ (-8,-4),(4,2) $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through the points (-8,-4) and (4,2) is \(y = 1/2x\).

Step 1: Calculating the slope

The slope \(m\) is calculated using the formula \((y2 - y1) / (x2 - x1)\). Substituting the given points into this formula gives \((2 - (-4)) / (4 - (-8)) = 6/12 = 1/2\). So, the slope \(m\) is 1/2.

Step 2: Determining the y-intercept

Substitute the slope and one of the given points into the slope-intercept equation and solve for \(b\). This gives, \(2 = 1/2 * 4 + b\). Solving for \(b\) yields \(b = 0\).

Step 3: Writing the final equation

Substitute the calculated slope and y-intercept into the slope-intercept equation to get the final equation of the line, \(y = 1/2x + 0\). Since adding 0 to anything doesn't change its value, this can be simplified to \(y = 1/2x\).

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and coordinate geometry. They represent straight lines on a graph, and their general form is usually given as \(y = mx + b\), known as the slope-intercept form. Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept where the line crosses the y-axis.
The slope indicates the degree of inclination or steepness of the line. It is calculated as the "rise over run," showing how much the line rises or falls as it moves from left to right.
The y-intercept is the value of \(y\) when \(x\) is zero, and it provides a starting point for graphing the line. By knowing these two components—slope and y-intercept—you can graph or find the equation of a line that represents various situations in algebra problems.

Coordinate Geometry

Coordinate geometry connects algebra and geometry by using coordinates to describe geometric figures, specifically lines. In a Cartesian coordinate system, points are assigned ordered pairs \((x, y)\), which correspond to positions on the grid.

• The x-coordinate indicates horizontal position.
• The y-coordinate shows vertical position.

To find the equation of a line that passes through given points, like \((-8, -4)\) and \((4, 2)\), you calculate the slope and use one of the points with the slope to find the y-intercept. Together, these give you the line's equation.
Coordinate geometry makes it easy to transform algebraic equations into visual graphs, helping to visualize the relationships between different algebraic expressions.

Algebra Problems

Algebra problems often require finding unknowns, such as solving for the slope and y-intercept in linear equations, which are common tasks in solving algebra problems.
These problems rely on applying formulas, such as the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of two points on the line. Once the slope \(m\) is determined, you substitute it into the slope-intercept form \(y = mx + b\) to find the y-intercept \(b\) using one of the points.
Solving algebra problems involving linear equations bridges quantitative reasoning with practical application, allowing students to tackle real-world situations like determining rates of change, predicting future trends, and graphing relationships.