Which of the following is an equation of a line with slope 5 and y-intercept 2?
The correct answer and explanation is:
To find the equation of a line with a slope of 5 and a y-intercept of 2, we use the slope-intercept form of a linear equation: y=mx+by = mx + b
Where:
- mm is the slope of the line
- bb is the y-intercept, which is the value of yy when x=0x = 0
Given:
- Slope m=5m = 5
- Y-intercept b=2b = 2
Substitute these values into the slope-intercept form: y=5x+2y = 5x + 2
✅ Correct Answer:
y=5x+2y = 5x + 2
✍️ Explanation (300 words):
In algebra, the equation of a straight line can be written in several forms, but the most common and easiest to interpret is the slope-intercept form, which is written as: y=mx+by = mx + b
This form is helpful because it gives us two key pieces of information immediately:
- The slope of the line, mm, which tells us how steep the line is or how much yy increases (or decreases) as xx increases by 1.
- The y-intercept, bb, which is the point where the line crosses the y-axis (i.e., where x=0x = 0).
In the problem, we are told that the slope of the line is 5. This means for every increase of 1 in xx, the value of yy increases by 5. Also, the y-intercept is 2, so the line crosses the y-axis at the point (0, 2).
Substituting these values into the slope-intercept form gives: y=5x+2y = 5x + 2
This equation means the line rises quickly due to its steep slope of 5 and starts at the y-axis at 2. It’s a linear relationship between xx and yy, showing that for every unit increase in xx, yy increases by 5 units from the base of 2.
Understanding this form is foundational for solving graphing problems, analyzing data, and modeling real-world relationships in algebra.