Problem 54 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line that passes through the given point and has the given slope, or that passes through the given points. \((4,-5)\) and \((-1,-3)\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through the points (4,-5) and (-1,-3) is y = -0.4x - 3.4

Step 1: Calculate the slope of the line

The formula used to calculate the slope (m) is given by (y2-y1)/(x2-x1). Plugging in the coordinates of the two points ((4,-5) and (-1,-3)), the slope, m = (-3-(-5))/(-1-4) = 2/(-5) = -0.4.

Step 2: Calculate the y-intercept

The slope-intercept equation is y = mx + b, where m is the slope, y and x are the coordinates of a point on the line, and b is the y-intercept. Solving for b using the point (4,-5), we get: -5 = -0.4*4 + b ==>> b = -5+1.6 = -3.4

Step 3: Write the equation in slope-intercept form

After having calculated the slope m and the y-intercept b, substitute these values into the simple linear equation form y = mx+b to write the equation in slope-intercept form. Thus, the equation becomes, y = -0.4x - 3.4

Key Concepts

Understanding Linear Equations

A linear equation represents a straight line on a coordinate plane. The general form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept. Linear equations are fundamental in mathematics as they express a relationship between two variables, \( x \) and \( y \), in which the rate of change between the variables is constant. This makes it a powerful tool to predict or understand behavior between variables.
When you solve a linear equation, you're really identifying all the points that satisfy this equation, forming a line. In practical terms, you are often given either a point and a slope or two points to find the equation in slope-intercept form, which is easy to graph and interpret.

How to Calculate the Slope

The slope of a line is a measure of its steepness and direction. Mathematically, it is the ratio of the vertical change (change in \( y \)) to the horizontal change (change in \( x \)) between two points on a line. The formula to calculate the slope \( m \) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Given two points, such as \((4, -5)\) and \((-1, -3)\), substitute these into the formula:

• \( m = \frac{-3 - (-5)}{-1 - 4} = \frac{2}{-5} = -0.4 \)

The negative sign indicates the line is decreasing or going downwards as it moves from left to right. Understanding the slope helps in predicting how the line behaves.

Finding the Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), \( b \) is the y-intercept. To find this, you can use the known slope and one point through which the line passes.
Using the slope calculated earlier (\( -0.4 \)) and the given point, such as \((4, -5)\), substitute into the linear equation:

• \(-5 = -0.4 \times 4 + b \)
• Solve for \( b \): \(-5 = -1.6 + b \)
• \( b = -5 + 1.6 = -3.4 \)

The y-intercept \( b = -3.4 \) indicates where the line crosses the y-axis in the coordinate plane. With both the slope and y-intercept, you can write the full equation of the line: \( y = -0.4x - 3.4 \). This equation shows exactly how the line behaves and where it starts on the graph.