Question

Write an equation of the line that passes through the points. $$(-3,12),(0,6)$$

Short answer

The equation of the line that passes through the given points (-3,12),(0,6) is \(y = -2x + 6\).

Step-by-step solution

Find the slope

To calculate the slope, subtract the y-coordinates of the two points and divide by the difference of the x-coordinates. \(m = \frac{12 - 6}{-3 - 0} = -2\).

Use the slope-intercept form of an equation

We plug one of the given points and the slope into the slope-intercept form of an equation, \(y = mx + b\), and solve for \(b\). Using the point (0,6), the equation becomes \(6 = -2*0 + b\). Solving for \(b\) gives \(b = 6\).

Write the final equation

Substitute the values of \(m\) and \(b\) into the equation \(y = mx + b\). This leads to the final equation: \(y = -2x + 6\).

Key concepts

Understanding Slope

The slope of a line is a measure that describes the direction and steepness of the line. It is calculated by finding the ratio of the vertical change to the horizontal change between two points on a line. In simpler terms, when you have two points on a line, the slope tells you how much the line increases or decreases as you move from one point to another.

For example, if the line moves up as it goes from left to right, the slope is positive. Conversely, if the line moves down, the slope is negative. The formula to find the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

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

In the exercise, with points (-3,12) and (0,6), the slope \(m\) is calculated as \( \frac{12 - 6}{-3 - 0} = -2\). This negative slope indicates that the line decreases as it goes from left to right.

Exploring Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line so that both the slope and the y-intercept are immediately visible. This form is useful for quickly understanding the properties of the line.

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 (the y-coordinate where the line crosses the y-axis). In the given solution, after determining the slope as \( -2 \), the next step was to solve for \( b \,\) the y-intercept, using one of the points provided.

Substituting the point (0,6) into the equation \( y = -2x + b \), we find that \( 6 = -2 \times 0 + b \), leading to \( b = 6 \). This tells us that the line crosses the y-axis at 6.

Fundamentals of Linear Equations

Linear equations represent straight lines and are fundamental in algebra. They form the basis of many real-life problem-solving situations.

A linear equation in two variables is usually written in the standard form as \( Ax + By = C \), where \( A, B, \) and \( C \) are constants. However, for simplicity, the slope-intercept form \( y = mx + b \) is often used to highlight the line's slope and intercept directly.

In the exercise, we found the equation \( y = -2x + 6 \). This is a linear equation representing the line that goes through the points (-3,12) and (0,6). It simplifies the task of graphing the line or predicting values, by clearly showing that for every unit increase in \( x \), the \( y \) value decreases by 2, starting from 6 when \( x = 0 \).

• Linear equations help predict unknown values and understand the relationship between variables.

This fundamental understanding is crucial for many applications in mathematics and real-life scenarios.