Question

Graph the line that contains the point P and has slope \(\mathrm{m}\). $$ P=(2,1) ; m=4 $$

Short answer

The line passing through (2,1) with slope 4 is y = 4x - 7.

Step-by-step solution

Determine the Equation of the Line

Use the point-slope formula to find the equation of the line. The point-slope formula is y - y_1 = m(x - x_1), where (x_1, y_1) is the given point and m is the slope. Plugging in P = (2,1) and m = 4: y - 1 = 4(x - 2).

Simplify the Equation

Distribute the slope and simplify the equation. Starting with: y - 1 = 4(x - 2), distribute 4 to get y - 1 = 4x - 8. Then, add 1 to both sides: y = 4x - 7.

Plot the Given Point

On a coordinate plane, locate and plot the point P = (2, 1).

Draw the Line

Using the slope m = 4, from point P (2,1), rise 4 units up and run 1 unit right to locate another point on the line, which would be at (3, 5). Plot this point as well. Draw the line through both points (2,1) and (3,5). This effectively plots the line y = 4x - 7.

Key concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most straightforward ways to represent a line. The general format for this form is y = mx + b. In this equation, 'm' represents the slope of the line and 'b' is the y-intercept. The y-intercept is the point where the line crosses the y-axis.

For example, if we have the equation y = 4x - 7, the slope 'm' is 4, indicating that for every unit we move right along the x-axis, we rise 4 units up. The intercept 'b' is -7, meaning the line crosses the y-axis at -7.

Using slope-intercept form makes it easy to graph a line because you can start at the y-intercept and use the slope to determine other points along the line. This form is particularly useful in quickly identifying the characteristics of the line.

Point-Slope Form

The point-slope form of a linear equation provides another way to write the equation of a line, especially when a point on the line and the slope are known. The format for this is: y - y_1 = m(x - x_1), where (x_1, y_1) represents a point on the line and 'm' is the slope.

To find the equation using the point-slope form for our exercise, plug in the given point P = (2,1) and slope m = 4. So, we get y - 1 = 4(x - 2). This formula highlights how the line passes through the specific point and moves according to the given slope.

The point-slope form is useful when you need to formulate an equation from given data and can easily be converted into other forms, like slope-intercept, for simpler graphing.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graph points, lines, and curves with the help of a horizontal axis (x-axis) and a vertical axis (y-axis). Each point on a coordinate plane is defined by an ordered pair (x, y), where x represents the horizontal distance from the origin, and y represents the vertical distance.

For instance, in the exercise, the point P = (2, 1) is located by moving 2 units right along the x-axis and 1 unit up along the y-axis. The coordinate plane helps in visualizing linear equations by plotting points and drawing lines through these points.

Our line from the example, y = 4x - 7, is graphed by first plotting the point (2, 1) and then using the slope to determine another point like (3, 5). Connecting these points with a straight line helps us see how the equation represents the line in the coordinate system.