Question

Write an equation of the line in slope-intercept form that passes through the two points, or passes through the point and has the given slope. $$(-2,4), m=3$$

Short answer

The equation of the line in slope-intercept form that passes through the points (-2,4) and has the given slope 3 is \(y = 3x + 10\).

Step-by-step solution

Identify the given slope and point

In the problem, it is given that the slope \(m\) is 3 and the line passes through the point (-2,4). So, \(m = 3\) and \(x_1 = -2\) and \(y_1 = 4\).

Substitute the known values into the formula to solve for \(b\)

The formula to find the y-intercept \(b\) when you have the slope and one point \(x_1, y_1\) is: \(b = y_1 - m*x_1\). Substituting the given values, we get: \(b = 4 - 3*(-2) = 4 + 6 = 10\).

Write the final slope-intercept form of the line

Upon substituting the value of slope \(m = 3\) and y-intercept \(b = 10\) into the slope-intercept form equation \(y = mx + b\), the equation of the line becomes \(y = 3x + 10\).

Key concepts

Writing Linear Equations

Understanding how to write linear equations is fundamental in algebra, especially in coordinate geometry. The most common form is the slope-intercept form, which is expressed as \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept.

To write an equation, you need to know at least the slope and one point, or two points through which the line passes. Once you have these, you plug the values into the slope-intercept form. If only a point and the slope are given, such as the point (-2,4) with a slope of 3, you directly substitute the slope in place of \(m\), and use the point to solve for \(b\), the y-intercept, as shown in the provided solution steps.

Slope of a Line

The slope of a line is a measure of its steepness or inclination and is a crucial concept in coordinate geometry. It is indicated by the letter \(m\) and calculated as the 'rise over run', or the change in y over the change in x between two points on the line.

If you have two points, \(x_1, y_1\) and \(x_2, y_2\), the slope is found using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). A positive slope means the line is rising as it moves from left to right, a negative slope means it's falling, and a slope of zero indicates a horizontal line. In the given example, the slope is 3, meaning for every unit the line moves horizontally, it rises by 3 units vertically.

Y-Intercept

The y-intercept of a line is the point where it crosses the y-axis, and it's always expressed as an ordered pair with zero as the x-coordinate, like \(0, b\).

In the slope-intercept form equation \(y = mx + b\), \(b\) denotes the y-intercept. To find \(b\) when you have a slope and a point, use the formula \(b = y_1 - m*x_1\), as done in the second step of our solution. It's pivotal because it helps to anchor the line in place on a graph. In our example, after working out the math, we find the y-intercept is 10, giving us the coordinate \(0, 10\) where our line intercepts the y-axis.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, lets us describe geometric shapes algebraically using coordinates and equations. It's a way of combining algebra and geometry to solve problems involving lines, angles, and curves.

In the context of linear equations, coordinate geometry helps you visualize the slope and y-intercept as parts of a line in a two-dimensional space. With the equation of the line \(y = 3x + 10\), you can plot it on a graph using the slope and y-intercept. This method is used to solve various real-world problems, like predicting trends or determining the distance between two points.