a. Sam knows that the x-intercept of a linear equation is 2 and that the y-intercept is 5. Graph this line on the coordinate plane.
b. Look at the graph and find the slope.
c. Write an equation of the line in slope-intercept form.
d. Use the x-intercept and the slope to write an equation in point-slope for the line.
e. Solve the equation in part d for y. Do you get the same equation that you had in part c? Will this always work? Why or why not?
a. To graph the line with x-intercept at 2 and y-intercept at 5, we can plot these two points on the coordinate plane and draw a line through them:
x-intercept: (2, 0)
y-intercept: (0, 5)
Now, drawing a line through these points:
b. To find the slope, we can use the formula:
slope = (y2 - y1) / (x2 - x1)
Using the points (2, 0) and (0, 5):
slope = (5 - 0) / (0 - 2) = 5 / -2 = -2.5
c. 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. Substituting the values we have:
y = -2.5x + 5
d. In point-slope form, the equation of a line is y - y1 = m(x - x1). Using the point (2, 0) and the slope -2.5:
y - 0 = -2.5(x - 2)
y = -2.5x + 5
e. Solving the equation in part d for y, we get y = -2.5x + 5, which is the same as the equation in part c. This method will always work because the slope-intercept form and the point-slope form are both equations of a line and are equivalent.