Question

Write an equation in slope-intercept form of the line that passes through the point and has the given slope. $$ (3,1), m=7 $$

Short answer

The equation of the line in slope-intercept form that passes through the point (3,1) with slope 7 is \(y = 7x - 20\).

Step-by-step solution

Know the Slope-Intercept Form

Understand that the slope-intercept form of the equation of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Substitute for \(m\)

Substitute \(m = 7\) into the equation. The equation becomes \(y = 7x + b\).

Substitute the given point

The given point (3,1) is a specific (x,y) pair that satisfies the equation. Substitute these values into the equation, so you get \(1 = 7*3 + b\).

Solve for \(b\)

Solve the equation from the previous step for \(b\). This will give you \(1 = 21 + b\), which simplifies to \(b = -20\).

Write the Final Equation

Substitute \(b = -20\) back into the equation \(y = 7x + b\) to give the final equation. This results in \(y = 7x - 20\). This is the equation of the line in slope-intercept form that passes through the point (3,1) with slope 7.

Key concepts

Understanding Linear Equations

A linear equation is one of the most fundamental concepts in algebra. It represents a straight line when plotted on a coordinate graph. Every linear equation can be written in the form of y = mx + b, which is known as the slope-intercept form.

In this form, m represents the slope of the line, which indicates the steepness and the direction of the line. The b represents the y-intercept, which is the point where the line crosses the y-axis. By knowing just the slope and y-intercept, we can graph the entire line.

Linear equations are incredibly useful because they can model real-world situations with a constant rate of change, such as speed, or to predict trends, like predicting sales over time. When working with linear equations, it is essential to understand how to manipulate the equation to find the slope and y-intercept, as these will be instrumental in graphing and solving problems connected to the line.

The Slope of a Line

The slope is a measure of how steep a line is in relation to the horizontal axis. Mathematically, it is calculated as the 'rise over run', which means the change in the y-values divided by the change in the x-values between two distinct points on the line.

The formula for slope, represented as m, is given as: \[ m = \frac{\text{change in y}}{\text{change in x}} = \frac{y_2 - y_1}{x_2 - x_1} \] A positive slope means that the line rises from left to right, a negative slope means that it falls, a zero slope means the line is horizontal, and an undefined slope means the line is vertical.

Understanding the slope is vital because it can tell us not only the direction of the line but also how one variable changes with respect to the other. For example, in the context of speed, a steeper slope (larger slope value) indicates a faster speed.

The Concept of y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form of a linear equation, y = mx + b, the y-intercept is represented by the constant b.

Identifying the y-intercept is straightforward because it's the value of y when x is zero. In other words, it's the starting point of the line when graphed on a coordinate plane. This value is significant as it can represent the starting quantity or initial value in real-world problems.

For instance, if the linear equation represents a business's profit over time, the y-intercept could represent the initial capital before any business activity begins. Knowing how to solve for and interpret the y-intercept is crucial in understanding the complete behavior of a line.