Question

Write an equation in slope-intercept form of the line with the given slope that passes through the given point. $$m=-2,(-2,1)$$

Short answer

The equation is \(y = -2x - 3\).

Step-by-step solution

Understanding Slope-Intercept Form

The slope-intercept form of a linear equation is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Our task is to find \(b\) given a point on the line and the slope.

Identify the Given Values

We are given the slope \(m = -2\) and a point \((-2, 1)\). This point means when \(x = -2\), \(y = 1\).

Substitute into the Slope-Intercept Equation

Substitute the slope \(m = -2\), \(x = -2\), and \(y = 1\) into the equation \(y = mx + b\). This leads to the equation: \[1 = -2(-2) + b\]

Solve for the Y-Intercept

Now simplify and solve for \(b\) in the equation: \[1 = 4 + b\]. Subtract 4 from both sides to get \(b = -3\).

Write the Equation

With \(m = -2\) and \(b = -3\), the equation in slope-intercept form is \(y = -2x - 3\).

Key concepts

Linear Equations

Linear equations are a foundational concept in algebra. These are equations that describe a straight line graphed on a coordinate plane. In general, a linear equation can be written in the form \(y = mx + b\). This form is known as the slope-intercept form and it's particularly helpful because it directly shows the slope and y-intercept of the line.
This makes it easy to graph the equation or understand the line's behavior.

• **Slope-intercept form**: A common way to express linear equations that highlights key properties.
• **Straight line**: The graph of a linear equation. It has constant rate of change, meaning that the slope is constant.

A linear equation isn't just limited to math classes. It's used in science, economics, and engineering to model relationships between two variables.

Slope

The slope of a line is a measure of its steepness. Often represented by the letter \(m\), the slope is calculated as the "rise" over the "run," or the change in \(y\) divided by the change in \(x\).
A positive slope means the line inclines upwards as it moves from left to right. A negative slope means it slopes downwards. In our example, the given slope is \(-2\). This means that for each unit the line moves to the right, it moves two units down.

• **Positive slope**: Line rises from left to right.
• **Negative slope**: Line falls from left to right, as in the example with \(-2\).
• **Zero slope**: A horizontal line, indicating no steepness.

Understanding slope is crucial because it tells us about how fast \(y\) changes with respect to \(x\). It's a key part of interpreting and working with linear equations.

Y-Intercept

The y-intercept is a specific point on the graph where the line crosses the y-axis. In slope-intercept form \(\left(y = mx + b\right)\), \(b\) represents the y-intercept. This is the value of \(y\) when \(x\) is zero. Knowing the y-intercept helps in quickly graphing a line because it's one of the few points you can easily determine without calculation.
In the given problem, we calculated the y-intercept to be \(-3\), which tells us that the line crosses the y-axis at the point \( (0, -3) \).

• **Point on y-axis**: The value of \(b\) in the equation gives us this point.
• **Quick graphing**: Starting point for drawing the line.
• **Real-world context**: Often represents an initial condition or starting point in applications.

The y-intercept is not just a technical detail—it's a fundamental part of what defines the position of a line within a graph.