Question

Write an equation in slope-intercept form of the line that passes through the point and has the given slope. $$ (6,5), m=2 $$

Short answer

The equation of the line in slope-intercept form that passes through the point (6,5) and has a slope of 2 is \(y = 2x - 7\)

Step-by-step solution

Understanding the given point and slope

The point given is \((6,5)\), meaning that when \(x=6\), \(y=5\). The slope, represented by \(m\), is 2. The slope-intercept form of the equation will then be in the form \(y = 2x + c\)

Substitute the x and y values into the equation

The x and y values from the given point can now be substituted into that equation to solve for \(c\). So \(5 = 2 * 6 + c\)

Solve for the y-intercept, c

Solve the above equation for \(c\): \(5 - 12 = c\). So \(c = -7\)

Write final equation

Now, substitute \(c = -7\) into the equation \(y = 2x + c\) to get the final equation: \(y = 2x - 7\)

Key concepts

Linear Equations

Linear equations are the simplest type of equations in algebra that describe a straight line when plotted on a graph. They have one or two variables, usually represented by x and y, and can be expressed in various formats. The most common form is the slope-intercept form, which we will explore in this discussion.

The general structure of a linear equation in two variables is \(ax + by = c\), where a, b, and c are constants. Understanding linear equations is crucial as they are not only the foundation for algebra but also applicable across various scientific fields.

Equation of a Line

The equation of a line provides the relationship between the x and y coordinates of any point on that line. The most common way to write the equation of a line is in the slope-intercept form, which is \(y = mx + b\). In this equation, m represents the slope of the line, while b is the y-intercept, the point where the line crosses the y-axis.

To find the equation of a line, one needs either two points on the line or one point and the slope. If the slope is unknown, it can be calculated using the coordinates of two given points on the line. Once the slope is known, along with one point, the slope-intercept form can be used to express the line's equation fully.

Slope

The slope of a line is a measure of its steepness and direction, calculated as the ratio of the change in y (the vertical change) to the change in x (the horizontal change) between two distinct points on the line. Represented by the letter \(m\), the slope is found using the formula \(m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\), where \(\Delta\) stands for 'change in.'

A positive slope means the line is rising from left to right, while a negative slope indicates the line is falling. A zero slope implies the line is horizontal, and an undefined slope (or an infinite value) indicates a vertical line. In the context of the given exercise, the slope was provided as 2, implying the line rises by 2 units vertically for every 1 unit it moves horizontally.

Y-intercept

The y-intercept is a specific point where the line crosses the y-axis on a graph. For the slope-intercept form of the linear equation \(y = mx + b\), the y-intercept is represented by \(b\). This value tells us the y coordinate of the point through which the line travels when x equals zero. It's essential when graphing the line, as it serves as a starting point on the y-axis from which we can use the slope to find other points on the line.

In our exercise, after determining the line's slope, we used the coordinates of the given point and the slope to find the y-intercept, which turned out to be -7. This means the line crosses the y-axis at the point \(0, -7\). As a result, the complete slope-intercept form of the equation is \(y = 2x - 7\), telling us that the line rises by 2 units for every horizontal unit it moves, starting from the y-intercept of -7.