GED Math: Lines and Slope
In real situations, slope is a rate of change: the amount of the y-quantity per one unit of the x-quantity (dollars per gallon, miles per hour, calls per day). When the slope is a whole number or decimal it is called the unit rate. To find it from a graph or table, pick two points and use m = (y2 − y1)/(x2 − x1), then divide to get a decimal or whole number. If the line passes through (0, 0), you can simply divide y by x for any point. A negative rate means the quantity is decreasing, for example a machine losing value. Always read the axis labels so that the unit of the answer is “y per x”, and match the order to the question: the rate of time per area is hours per square yard, not the reverse.
1. Graphing a Line and Checking Points
A linear equation has two variables (usually x and y), no exponents on them, and its graph is a straight line. Every point on the line is a solution, so a line has infinitely many solutions, but you only need two points to draw it (a third is a good check). To graph an equation: (1) pick several values for x, (2) use the equation to find each matching y, (3) record the pairs in a table of values, (4) plot the points and (5) draw a straight line through them with arrows on both ends. If y is not alone, use inverse operations to isolate it first: x + y = 7 becomes y = −x + 7. To test whether a point is on a line, substitute its x and y into the equation; the point is on the line only if the equation stays true.
Worked Example: Graph y = 2x − 3.
Choose x = 0, 1, 2, 3. Then y = −3, −1, 1, 3. Plot (0, −3), (1, −1), (2, 1), (3, 3) and draw the line through them.
1. Complete the table for y = 2x − 3. What is y when x = 5?
Explanation: y = 2(5) − 3 = 10 − 3 = 7.
2. For y = −3x + 5, what is y when x = 2?
Explanation: y = −3(2) + 5 = −6 + 5 = −1.
3. For y = 12x + 3, what is y when x = −4?
Explanation: y = ½(−4) + 3 = −2 + 3 = 1.
4. Solve x + y = 7 for y.
Explanation: Subtract x from both sides: y = −x + 7.
5. Solve 2x + y = 15 for y.
Explanation: Subtract 2x from both sides: y = −2x + 15.
6. Solve y − 4 = 3x for y.
Explanation: Add 4 to both sides: y = 3x + 4.
7. Which point lies on the line y = 3x − 4?
Explanation: At x = 2: y = 3(2) − 4 = 2, so (2, 2) works. The others fail the equation.
8. Which point lies on the line y = −x + 6?
Explanation: At x = 4: y = −4 + 6 = 2, so (4, 2) works.
9. Which equation describes a line that passes through the point (5, −6)?
Explanation: Substitute x = 5: y = −4x + 14 gives y = −4(5) + 14 = -6, which matches. The other equations give different y-values at x = 5.
10. Which equation describes a line that passes through the point (−2, 7)?
Explanation: Substitute x = -2: y = −3x + 1 gives y = −3((−2)) + 1 = 7, which matches. The other equations give different y-values at x = -2.
11. What is the fewest number of points needed to draw a straight line?
Explanation: Two points determine exactly one straight line (a third is only a check).
12. Which equation describes the line shown?
Explanation: The line passes through (0, −3) and (1, −1), so it is y = 2x − 3.
2. Slope as Rise Over Run
Slope measures how steep a line is. It is the ratio of the vertical change (the rise) to the horizontal change (the run) between any two points on the line: slope = rise / run. Moving from left to right, a line that goes up has a positive slope and a line that goes down has a negative slope. The larger the absolute value of the slope, the steeper the line: a slope of 3 is steeper than a slope of 12. To find the slope from a graph, choose two points where the line crosses grid intersections, count the rise and the run (up and right are positive, down and left are negative), and write rise over run in lowest terms. A line has the same slope everywhere.
Worked Example: Find the slope of the line through A (−2, −1) and B (2, 2).
From A to B the line rises 3 and runs 4, so the slope is 34.
1. What is the slope of the line shown?
Explanation: Rise = 3 and run = 4, so the slope is 3/4.
2. What is the slope of the line shown?
Explanation: The line falls 4 while running 6: slope = −4/6 = −2/3.
3. A line goes up from left to right. Its slope is:
Explanation: Lines that rise as you move right have a positive slope.
4. A line goes down from left to right. Its slope is:
Explanation: Lines that fall as you move right have a negative slope.
5. Which line is the steepest?
Explanation: The greatest absolute value gives the steepest line, so slope 4 is steepest.
6. A line rises 6 units for every 2 units it runs to the right. What is its slope?
Explanation: Slope = rise/run = 6/2 = 3.
7. A line falls 5 units for every 10 units it runs to the right. What is its slope?
Explanation: Slope = −5/10 = −1/2.
8. A ramp rises 2 feet over a horizontal distance of 24 feet. What is the slope of the ramp?
Explanation: Slope = rise/run = 2/24 = 1/12.
3. The Slope Formula, Horizontal and Vertical Lines
When you have two points but no graph, use the slope formula: m = (y2 − y1) / (x2 − x1). Subtract the y-values on top and the x-values on the bottom, and keep the same order in both: if you start with the first point's y on top, start with its x on the bottom. Subtract carefully with negatives. Two special cases are worth memorizing. A horizontal line (same y everywhere) has a rise of 0, so its slope is 0. A vertical line (same x everywhere) has a run of 0, so its slope is undefined. Test tip: the formula is on the GED formula sheet, but you still need to substitute carefully.
Worked Example: Find the slope of the line through (3, −2) and (−1, 6).
m = (6 − (−2)) / (−1 − 3) = 8 / (−4) = −2.
1. What is the slope of the line through (3, 5) and (−1, 2)?
Explanation: m = (y2 − y1) / (x2 − x1) = (2 − 5) / (−1 − 3) = −3/−4 = 34.
2. What is the slope of the line through (0, 2) and (4, 0)?
Explanation: m = (y2 − y1) / (x2 − x1) = (0 − 2) / (4 − 0) = −2/4 = −12.
3. What is the slope of the line through (6, 1) and (0, 3)?
Explanation: m = (y2 − y1) / (x2 − x1) = (3 − 1) / (0 − 6) = 2/−6 = −13.
4. What is the slope of the line through (−2, −3) and (4, 5)?
Explanation: m = (y2 − y1) / (x2 − x1) = (5 − (−3)) / (4 − (−2)) = 8/6 = 43.
5. What is the slope of the line through (5, −2) and (2, −8)?
Explanation: m = (y2 − y1) / (x2 − x1) = (−8 − (−2)) / (2 − 5) = −6/−3 = 2.
6. What is the slope of the line through (4, 2) and (−3, 2)?
Explanation: Both points have y = 2, so the rise is 0 and the slope is 0 (a horizontal line).
7. What is the slope of the line through (5, 1) and (5, −4)?
Explanation: Both points have x = 5, so the run is 0. Division by 0 is undefined (a vertical line).
8. Which line has an undefined slope?
Explanation: The line x = 7 is vertical, so its slope is undefined.
9. The line through (1, 2) and (5, k) has slope 12. What is k?
Explanation: (k − 2)/(5 − 1) = 1/2, so k − 2 = 2 and k = 4.
10. The line through (2, 3) and (6, 11) has slope m. Another point on this line is:
Explanation: m = 8/4 = 2. From (2, 3), moving 2 right raises y by 4, so (4, 7) is on the line.
4. The Slope-Intercept Form: y = mx + b
The slope-intercept form of a line is y = mx + b. Here m is the slope and b is the y-intercept, the y-value where the line crosses the y-axis (the point (0, b)). The letters x and y stay as variables for any point on the line. To write an equation from a slope and a point: (1) substitute the slope and the point's coordinates for m, x, and y; (2) solve for b using inverse operations; (3) rewrite the equation with m and b filled in. To read a graph, find the y-intercept first, then count rise over run to get the slope. If an equation is not in this form, solve for y first.
Worked Example: Write the equation of the line with slope 4 that passes through (−1, −6).
Substitute: −6 = 4(−1) + b, so −6 = −4 + b and b = −2. The equation is y = 4x − 2.
1. What are the slope and y-intercept of y = 5x − 8?
Explanation: In y = mx + b, m = 5 and b = −8.
2. What is the slope of the line y = −23x + 1?
Explanation: The coefficient of x is the slope: −2/3.
3. What is the y-intercept of 3x + y = 7?
Explanation: Solve for y: y = −3x + 7. The y-intercept is 7.
4. What is the slope of the line 2x + y = 9?
Explanation: Solve for y: y = −2x + 9. The slope is −2.
5. Which equation describes the line with slope 3 that passes through (2, 1)?
Explanation: Substitute into y = mx + b: 1 = 3(2) + b, so b = 1 − (3 × 2) = −5. The equation is y = 3x − 5.
6. Which equation describes the line with slope 2 that passes through (−1, −4)?
Explanation: Substitute into y = mx + b: -4 = 2(-1) + b, so b = -4 − (2 × -1) = −2. The equation is y = 2x − 2.
7. Which equation describes the line with slope −13 that passes through (−4, 2)?
Explanation: Substitute into y = mx + b: 2 = −13(-4) + b, so b = 2 − (-13 × -4) = 23. The equation is y = −13x + 23.
8. Which equation describes the line with slope −2 that passes through (3, 5)?
Explanation: Substitute into y = mx + b: 5 = −2(3) + b, so b = 5 − (-2 × 3) = 11. The equation is y = −2x + 11.
9. Which equation describes the line shown?
Explanation: The y-intercept is −2. From (0, −2) the line rises 3 for every 2 to the right, so m = 3/2.
10. Which equation describes the line shown?
Explanation: The y-intercept is 3. Going from (0, 3) to (3, 1) falls 2 over a run of 3, so m = −2/3.
5. The Point-Slope Form and Equations from Two Points
The second form is the point-slope form: y − y1 = m(x − x1). Here (x1, y1) is a known point on the line and m is the slope. To use it: (1) call the given point (x1, y1); (2) substitute those two values; (3) substitute the slope for m. If the point has a negative coordinate, subtracting it becomes addition: with (x1, y1) = (−3, 2) you get y − 2 = m(x + 3). To convert to slope-intercept form, distribute and solve for y. To find an equation from two points, first compute the slope with the slope formula, then plug the slope and either point into the point-slope or slope-intercept form. Either point gives the same line.
Worked Example: Write the equation of the line through (4, 3) with slope 2.
Point-slope form: y − 3 = 2(x − 4). Distribute: y − 3 = 2x − 8. Add 3: y = 2x − 5.
1. Which equation is the point-slope form of the line with slope 3 through (2, 1)?
Explanation: Use y − y1 = m(x − x1) with (x1, y1) = (2, 1): y − 1 = 3(x − 2).
2. Which equation is the point-slope form of the line with slope −2 through (−3, 5)?
Explanation: x1 = −3, so x − (−3) = x + 3; y1 = 5. The slope is −2: y − 5 = −2(x + 3).
3. Write y − 3 = 2(x − 5) in slope-intercept form.
Explanation: Distribute: y − 3 = 2x − 10. Add 3: y = 2x − 7.
4. Which equation describes the same line as y − 2 = 12(x − 6)?
Explanation: Distribute: y − 2 = x/2 − 3. Add 2: y = x/2 − 1.
5. Write y + 4 = −3(x − 1) in slope-intercept form.
Explanation: Distribute: y + 4 = −3x + 3. Subtract 4: y = −3x − 1.
6. Which equation describes the line through (−5, 3) and (1, 1)?
Explanation: Slope: (1 − 3) / (1 − (−5)) = −13. Use point (−5, 3): b = 3 − (-13)((−5)) = 43. The line is y = −13x + 43.
7. Which equation describes the line through (−3, 0) and (−2, 4)?
Explanation: Slope: (4 − 0) / (-2 − (−3)) = 4. Use point (−3, 0): b = 0 − (4)((−3)) = 12. The line is y = 4x + 12.
8. Which equation describes the line through (−3, −4) and (7, 1)?
Explanation: Slope: (1 − (−4)) / (7 − (−3)) = 12. Use point (−3, −4): b = -4 − (12)((−3)) = −52. The line is y = 12x − 52.
9. Which equation describes the line through (−1, 0) and (2, −3)?
Explanation: Slope: (-3 − 0) / (2 − (−1)) = −1. Use point (−1, 0): b = 0 − (-1)((−1)) = −1. The line is y = −x − 1.
10. Which equation describes the line shown?
Explanation: From (0, 4) to (2, 0) the line falls 4 over a run of 2, so m = −2 and b = 4.
11. The line through (2, 0) has slope −13. Which point also lies on it?
Explanation: Moving 6 right changes y by −6/3 = −2, so from (2, 0) you reach (8, −2).
6. Slope as Unit Rate and Rate of Change
In real situations, slope is a rate of change: the amount of the y-quantity per one unit of the x-quantity (dollars per gallon, miles per hour, calls per day). When the slope is a whole number or decimal it is called the unit rate. To find it from a graph or table, pick two points and use m = (y2 − y1)/(x2 − x1), then divide to get a decimal or whole number. If the line passes through (0, 0), you can simply divide y by x for any point. A negative rate means the quantity is decreasing, for example a machine losing value. Always read the axis labels so that the unit of the answer is “y per x”, and match the order to the question: the rate of time per area is hours per square yard, not the reverse.
Worked Example: The graph shows the cost of fuel. Find the cost per gallon.
Use (2, 5) and (6, 15): (15 − 5)/(6 − 2) = 10/4 = $2.50 per gallon.
1. Using the graph, what is the cost of fuel per gallon?
Explanation: Rate = (15 − 5)/(6 − 2) = 10/4 = 2.5 dollars per gallon.
2. The graph shows the value of a machine. At what rate does its value change?
Explanation: Rate = (440 − 800)/(6 − 0) = −360/6 = −60. It loses $60 each month.
3. Using the graph, how much does the worker earn per hour?
Explanation: Rate = 120/10 = 12 dollars per hour (the line passes through the origin).
4. A caterer orders 150 lb of clams for 100 people and 30 lb for 20 people. How many pounds per person?
Explanation: Rate = (150 − 30)/(100 − 20) = 120/80 = 1.5 lb per person.
5. A painter uses 15 gallons to cover 300 sq yd and 8 gallons to cover 160 sq yd. How many gallons per square yard?
Explanation: Rate = (15 − 8)/(300 − 160) = 7/140 = 0.05 gallon per sq yd.
6. A scientist records (3, 2.4), (8, 6.4), and (9, 7.2). What is the rate of y per x?
Explanation: Rate = (6.4 − 2.4)/(8 − 3) = 4/5 = 0.8.
7. Oranges cost $0.44 per pound. Which point lies on the line of cost (y) versus pounds (x)?
Explanation: Cost = 0.44 × 5 = 2.20, so (5, 2.20) is on the line.
8. A record spins at 78 revolutions per minute. How many revolutions occur in 4 minutes?
Explanation: 78 × 4 = 312 revolutions.
9. Tank water drains from 90 gallons to 30 gallons in 6 minutes. What is the rate of change?
Explanation: Rate = (30 − 90)/6 = −60/6 = −10 gallons per minute.
10. On three carpet jobs, a contractor records (40, 1.5), (100, 2.25), and (300, 4.75) as (square yards, hours). What is her time per square yard, ignoring the 1 hour of travel?
Explanation: Rate = (2.25 − 1.5)/(100 − 40) = 0.75/60 = 0.0125 hr per sq yd.
Common Mistakes to Avoid
- Mixing up rise and run, giving run over rise. Rise (vertical) is always on top.
- Subtracting the y-values in one order and the x-values in the other, which flips the sign of the slope.
- Forgetting that subtracting a negative number means adding.
- Saying a vertical line has slope 0. Vertical lines have an undefined slope; horizontal lines have slope 0.
- Mixing up x and y when substituting a point into y = mx + b.
- Losing a sign when solving for b, for example writing b = y + mx instead of b = y − mx.
- Writing x + 3 as x − 3 in point-slope form. Subtracting −3 gives x + 3.
- Not rewriting y alone before reading the slope: in 2x + y = 9 the slope is −2, not 2.
Frequently Asked Questions
What is the slope of a line?
The ratio of rise to run between two points on the line. It tells how steep the line is and whether it goes up (positive) or down (negative).
What is the slope formula?
m = (y2 − y1) / (x2 − x1). Subtract in the same order on top and bottom.
What is the slope of a horizontal or vertical line?
A horizontal line has slope 0. A vertical line has an undefined slope.
What is slope-intercept form?
y = mx + b, where m is the slope and b is the y-intercept.
What is point-slope form?
y − y1 = m(x − x1), where (x1, y1) is a point on the line.
How do I find the equation of a line from two points?
Find the slope with the slope formula, then substitute the slope and one point into the point-slope or slope-intercept form.
How is slope related to rate?
In a real situation the slope is the rate of change: the amount of y per one unit of x.
Where can I practice more problems like these?
The GED Math practice in the GED Math QBank gives you more practice on this topic, and you can browse every other topic in the full GED Math QBank.