Question

Find an equation for the line with the given properties. Express your answer using either the general form or the slope-intercept form of the equation of a line, whichever you prefer. Slope \(=2\); containing the point (4,-3)

Short answer

The equation of the line is \(y = 2x - 11\).

Step-by-step solution

Understand the equation of a line

The slope-intercept form of the equation of a line is given by \(y = mx + b\). Here, \(m\) is the slope and \(b\) is the y-intercept.

Insert the given slope

Given the slope \(m = 2\), substitute it into the slope-intercept form of the equation: \(y = 2x + b\).

Use the given point to find the y-intercept

The line passes through the point (4, -3). Substitute \(x = 4\) and \(y = -3\) into the equation \(y = 2x + b\) to find \(b\).\(-3 = 2(4) + b\).\(-3 = 8 + b\)\ which simplifies to \(b = -11\).

Write the final equation

Now that \(b = -11\), substitute it back into the slope-intercept form equation: \(y = 2x - 11\).

Key concepts

Slope-Intercept Form

To start understanding linear equations, we need to know about the slope-intercept form. This form is written as \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. The slope-intercept form is useful because it easily shows us both the slope of the line and where it crosses the y-axis. This makes it simple to graph and analyze lines.
For example, in the problem given, the slope \(m\) is 2. Once we have the slope and at least one point that the line passes through, we can figure out the line's equation.

Finding Slope

The slope \(m\) of a line measures how steep the line is. It is calculated as the ratio of the change in the y-coordinates to the change in the x-coordinates between two points on the line. This is given by the formula: \(\frac{\text{change in y}}{\text{change in x}} = \frac{y_2 - y_1}{x_2 - x_1}\).
This slope tells us how the y value of a point on the line changes as the x value increases. A positive slope means the line rises from left to right, while a negative slope means the line falls. In the given problem, the slope is already provided as 2, indicating a rise of 2 units in y for every unit increase in x.

Y-Intercept Calculation

To find the y-intercept \(b\), we need a point on the line and the slope. Using the slope-intercept form \(y = mx + b\), we can substitute the x and y values of a given point and solve for \(b\).
In our exercise, the line passes through the point (4, -3) with a slope of 2. Plugging these values into the formula, we get: \(-3 = 2(4) + b\). Simplifying this, we find \(b\) by solving \(-3 = 8 + b\). Subtract 8 from both sides to get \(b = -11\).
So, the y-intercept here is -11, which means the line crosses the y-axis at \(y = -11\).

Linear Equations

A linear equation represents a straight line in the coordinate plane. It can be written in various forms, but the most common are the slope-intercept form \(y = mx + b\) and the general form \(Ax + By + C = 0\).
Linear equations are called 'linear' because their graphs are straight lines. Each pair of (x, y) values that solves the equation lies on this straight line.
In our example, after finding the slope and y-intercept, we write the final equation of the line: \(y = 2x - 11\). This means for any x value, you can find the corresponding y value by substituting x into the equation.
Understanding linear equations helps in graphing lines and interpreting relationships between variables in many real-world scenarios.