Slope from a graph, equation or table

1. Label the points: (x_1, y_1) = (1, 2) and (x_2, y_2) = (4, 8).
2. Use the slope formula: m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 2}{4 - 1} = \frac{6}{3}
3. Simplify: \dfrac{6}{3} = 2.

The slope is 2.

Check: Starting at (1, 2), each step of 1 to the right goes up 2: (2, 4), (3, 6), (4, 8) — we land exactly on the second point ✓

Tip: "Rise over run": the y-change (up/down) goes on top, the x-change (across) on the bottom.

1. Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.
2. Match y = -3x + 7 with y = mx + b: m = -3, \qquad b = 7
3. The slope is the number multiplying x, including its sign.

The slope is -3.

Check: Pick two x-values: x = 0 \Rightarrow y = 7; x = 1 \Rightarrow y = -3 + 7 = 4. Change in y is 4 - 7 = -3 for a change in x of 1, so m = \frac{-3}{1} = -3 ✓

Tip: The slope is the coefficient of x only when y is alone on one side of the equation.

1. Use two points that sit exactly on grid corners: (0, 0) and (2, 3).
2. Rise: from y = 0 up to y = 3 is +3.
3. Run: from x = 0 right to x = 2 is +2.
4. Slope: m = \frac{\text{rise}}{\text{run}} = \frac{3}{2}

The slope is \tfrac{3}{2} (or 1.5).

Check: Use the other two points (-2, -3) and (2, 3): \frac{3 - (-3)}{2 - (-2)} = \frac{6}{4} = \frac{3}{2} ✓. The line rises left to right, so a positive slope makes sense.

Tip: Any two points on a line give the same slope — choose ones where the line crosses a grid intersection exactly.

1. Label the points: (x_1, y_1) = (3, 1) and (x_2, y_2) = (7, 13).
2. Slope formula (on the GED formula sheet): m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{13 - 1}{7 - 3} = \frac{12}{4}
3. Simplify: \dfrac{12}{4} = 3.

The slope is 3.

Check: From (3, 1), go 1 right and 3 up four times: (4, 4), (5, 7), (6, 10), (7, 13) — we reach the second point ✓

Tip: The formula uses differences (subtraction), never sums. Rise (the y-change) on top, run (the x-change) on the bottom.

1. Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.
2. Reorder the right side so the x-term comes first (addition can be done in any order, but keep each term's sign with it): y = 9 - 4x = -4x + 9
3. Compare with y = mx + b: m = -4 and b = 9.

The slope is -4.

Check: Test two points: x = 0 \Rightarrow y = 9; x = 1 \Rightarrow y = 5. Slope = \frac{5 - 9}{1 - 0} = -4 ✓

Tip: The slope is the coefficient of x, wherever x appears. The sign in front of the term belongs to the coefficient: -4x means m = -4.

1. Subtract 4x from both sides: 2y = -4x + 10
2. Divide every term by 2: y = -2x + 5
3. Now the equation is in the form y = mx + b with m = -2.

The slope is -2.

Check: Find two points on the original line. x = 0: 2y = 10, y = 5 → (0, 5). x = 1: 4 + 2y = 10, y = 3 → (1, 3). Slope = \frac{3 - 5}{1 - 0} = -2 ✓

Tip: Shortcut for Ax + By = C: slope = -\dfrac{A}{B} = -\dfrac{4}{2} = -2. Never read the slope off an equation that isn't solved for y.

1. Use the marked points (0,\ 4) and (3,\ -2).
2. Rise: from y = 4 down to y = -2: -2 - 4 = -6
3. Run: from x = 0 to x = 3: 3 - 0 = 3
4. Slope: m = \frac{-6}{3} = -2

The slope is -2.

Check: The line also crosses (2, 0). From (0, 4) to (2, 0): \frac{0 - 4}{2 - 0} = \frac{-4}{2} = -2 ✓. The line falls from left to right, so a negative slope is right.

Tip: Read left to right like a sentence: uphill = positive slope, downhill = negative slope.

1. Let (x_1, y_1) = (-3, 5) and (x_2, y_2) = (1, -3).
2. Change in y: y_2 - y_1 = -3 - 5 = -8
3. Change in x: x_2 - x_1 = 1 - (-3) = 1 + 3 = 4
4. Slope: m = \frac{-8}{4} = -2

The slope is -2.

Check: Reverse the order: \frac{5 - (-3)}{-3 - 1} = \frac{8}{-4} = -2 ✓ — the same answer, as it must be.

Tip: Put parentheses around negative numbers before subtracting: 1 - (-3). Most slope errors are sign errors, not formula errors.

1. The marked points are (-4, -1) and (4, 5).
2. Rise (change in y): 5 - (-1) = 6.
3. Run (change in x): 4 - (-4) = 8.
4. Slope: m = \frac{\text{rise}}{\text{run}} = \frac{6}{8} = \frac{3}{4}
5. The line rises from left to right, so a positive slope makes sense.

The slope is \tfrac{3}{4}.

Check: From (4, 5), go back 4 left and 3 down: (0, 2). The graph crosses the y-axis at 2 ✓

Tip: Use points where the line crosses grid intersections exactly (the marked points). Always reduce the fraction and check the sign against the picture: rising = positive.

1. Both points have y = 6, so the line is horizontal (the line y = 6).
2. Slope formula: m = \frac{6 - 6}{5 - (-2)} = \frac{0}{7}
3. Zero divided by any nonzero number is 0.

The slope is 0.

Check: A horizontal line neither rises nor falls as you move right, so its rise is 0 and its slope is 0 ✓

Tip: \frac{0}{\text{number}} = 0 (horizontal line), but \frac{\text{number}}{0} is undefined (vertical line). Remember "Horizontal Zero, Vertical Undefined".

1. Label: (x_1, y_1) = (-4, -1), (x_2, y_2) = (2, -5).
2. Rise: y_2 - y_1 = -5 - (-1) = -5 + 1 = -4.
3. Run: x_2 - x_1 = 2 - (-4) = 2 + 4 = 6.
4. Slope: m = \frac{-4}{6} = -\frac{2}{3}

The slope is -\tfrac{2}{3}.

Check: Use the points in the other order: \frac{-1 - (-5)}{-4 - 2} = \frac{4}{-6} = -\frac{2}{3} — same answer ✓ Also, y goes down while x goes up, so the slope must be negative ✓

Tip: "Minus a negative" becomes "plus": -5 - (-1) = -5 + 1. Put negatives in parentheses in your work so you don't lose a sign.

1. Start with 3x - 5y = 15.
2. Subtract 3x from both sides: -5y = -3x + 15
3. Divide every term by -5: y = \frac{-3}{-5}x + \frac{15}{-5} = \frac{3}{5}x - 3
4. Compare with y = mx + b: m = \tfrac{3}{5}.

The slope is \tfrac{3}{5}.

Check: Two points on the line: x = 0 \Rightarrow -5y = 15 \Rightarrow y = -3; x = 5 \Rightarrow 15 - 5y = 15 \Rightarrow y = 0. Slope = \frac{0 - (-3)}{5 - 0} = \frac{3}{5} ✓

Tip: Shortcut for Ax + By = C: slope = -\dfrac{A}{B}. Here -\dfrac{3}{-5} = \dfrac{3}{5}. A negative divided by a negative is positive.

1. In the slope formula, the same point must come first on top and on bottom: \dfrac{y_2 - y_1}{x_2 - x_1}.
2. On top, Marcus used 5 - (-3): second point minus first point.
3. On the bottom, he used 2 - 6: first point minus second point. The order is mixed — this is the error, in Step 1.
4. Correct work: m = \frac{5 - (-3)}{6 - 2} = \frac{8}{4} = 2
5. Steps 2 and 3 are correct arithmetic for what he wrote; the mistake started in Step 1.

Step 1 is wrong; the slope is 2.

Check: As x goes from 2 to 6 (up), y goes from -3 to 5 (up), so the slope must be positive ✓

Tip: Mixed order always flips the sign. A quick "rising or falling?" sense check catches it.

Steepness is measured by the absolute value of the slope, \lvert m \rvert. The sign only tells direction.

1. y = -4x + 1: m = -4, so \lvert m \rvert = 4.
2. y = 3x + 9: m = 3, so \lvert m \rvert = 3.
3. Line through (0, 0) and (2, 7): m = \frac{7 - 0}{2 - 0} = 3.5, so \lvert m \rvert = 3.5.
4. 6x - 2y = 4: -2y = -6x + 4 \Rightarrow y = 3x - 2, so m = 3, \lvert m \rvert = 3.

Compare: 4 > 3.5 > 3. The steepest line is y = -4x + 1.

Check: For each 1 step right, y = -4x + 1 drops 4 units; no other line changes by more than 3.5 units per step ✓

Tip: A slope of -4 is steeper than a slope of 3. "Steepest" compares sizes; "greatest slope" compares signed values — read which one the question asks.

1. Slope formula with the given points: \frac{-10 - 2}{4 - k} = 4 \quad\Rightarrow\quad \frac{-12}{4 - k} = 4
2. Multiply both sides by (4 - k): -12 = 4(4 - k) = 16 - 4k
3. Subtract 16 from both sides: -28 = -4k.
4. Divide by -4: k = 7.

The value is k = 7.

Check: Points (7, 2) and (4, -10): m = \frac{-10 - 2}{4 - 7} = \frac{-12}{-3} = 4 ✓

Tip: In "find the missing coordinate" problems, plug the unknown into the slope formula and solve. Then always substitute back to check — sign errors are very common here.

1. Find each slope:
   • y = -\tfrac{1}{4}x + 3: m = -0.25, steepness \lvert m \rvert = 0.25.
   • Line through (0, 0) and (5, 2): m = \tfrac{2}{5} = 0.4, \lvert m \rvert = 0.4.
   • 2x + 4y = 9: 4y = -2x + 9, so y = -\tfrac{1}{2}x + \tfrac{9}{4}; m = -0.5, \lvert m \rvert = 0.5.
   • Line where y increases by 1 each time x increases by 3: m = \tfrac{1}{3} \approx 0.33, \lvert m \rvert \approx 0.33.
2. Compare steepness: 0.25 < 0.33 < 0.4 < 0.5.
3. The smallest absolute value is 0.25.

The least steep line is y = -\tfrac{1}{4}x + 3.

Check: Over a run of 12, the four lines change by 3, 4.8, 6 and 4 units vertically; the smallest change is 3, for m = -\tfrac{1}{4} ✓

Tip: Steepness = \lvert m \rvert. The smallest signed number (-0.5 here) is not the least steep — it is actually the steepest of these four.

1. To isolate 2y in 6x + 2y = 8, subtract 6x from both sides: 2y = 8 - 6x = -6x + 8 Rosa wrote +6x, so Step 1 contains the error.
2. Continue correctly: divide every term by 2: y = -3x + 4
3. The slope is the coefficient of x: m = -3.
4. Rosa's Step 2 divided both terms by 2 correctly, and her Step 3 read the coefficient correctly — but both were built on the sign error.

Step 1 is wrong; the slope is -3.

Check: Two points on 6x + 2y = 8: (0, 4) and (1, 1). Slope = \frac{1 - 4}{1 - 0} = -3 ✓

Tip: When a term moves across the equals sign, its sign flips (because you subtract it from both sides). Shortcut for Ax + By = C: m = -\frac{A}{B} = -\frac{6}{2} = -3.

1. A 6% grade means a slope of \frac{6}{100} = 0.06 (6 feet of drop per 100 feet across).
2. Since slope = \dfrac{\text{rise}}{\text{run}}, we have rise = slope \times run.
3. Multiply: \text{drop} = 0.06 \times 1{,}500 = 90 \text{ feet}

The road drops 90 feet.

Check: 1,500 feet is 15 groups of 100 feet; at 6 feet per 100 feet, 15 \times 6 = 90 feet ✓

Tip: A percent grade is just slope × 100. Convert the percent to a decimal (move the point two places left) before multiplying.

1. The marked points are (0, 400) and (8, 1600): x in miles, y in feet.
2. Rise: 1600 - 400 = 1200 feet.
3. Run: 8 - 0 = 8 miles.
4. Slope: m = \frac{1200 \text{ ft}}{8 \text{ mi}} = 150 \text{ feet per mile}

The slope is 150 feet per mile — the trail climbs 150 feet for each mile hiked.

Check: After 4 miles: 400 + 150 \times 4 = 1000 feet. The line passes through 1,000 at 4 miles on the graph ✓

Tip: When the axes have different scales (200 per square up, 2 per square across), counting squares gives the wrong slope. Always use the axis values of two points.

1. Rewrite each equation in y = mx + b form:
   • y = \dfrac{4 - x}{2} = \dfrac{4}{2} - \dfrac{x}{2} = -\tfrac{1}{2}x + 2 → m = -\tfrac{1}{2} ✓
   • y = -2x + \tfrac{1}{2} → m = -2.
   • y = \tfrac{1}{2} - x = -1x + \tfrac{1}{2} → m = -1.
   • y = \dfrac{x - 1}{2} = \tfrac{1}{2}x - \tfrac{1}{2} → m = \tfrac{1}{2}.
2. Only y = \dfrac{4 - x}{2} has m = -\tfrac{1}{2}.

The answer is y = \dfrac{4 - x}{2}.

Check: Points on y = \frac{4 - x}{2}: x = 0 \Rightarrow y = 2; x = 2 \Rightarrow y = 1. Slope = \frac{1 - 2}{2 - 0} = -\frac{1}{2} ✓

Tip: A fraction bar under a whole expression divides every term. The slope is the coefficient of x after splitting — -\frac{x}{2} means -\tfrac{1}{2}x.

1. Slope -\tfrac{4}{3} = rise -4 over run 3: from any point, move 3 right and 4 down to reach another point.
2. From (3, 2): (3 + 3,\ 2 - 4) = (6,\ -2)
3. Test the other choices with the slope formula from (3, 2):
   • (7, -1): \frac{-1 - 2}{7 - 3} = -\frac{3}{4} ✗
   • (0, -2): \frac{-2 - 2}{0 - 3} = \frac{4}{3} ✗
   • (6, 6): \frac{6 - 2}{6 - 3} = \frac{4}{3} ✗

The point (6, -2) is on the line.

Check: \frac{-2 - 2}{6 - 3} = \frac{-4}{3} ✓

Tip: For a negative slope, the two moves go in opposite directions (right & down, or left & up). Moving left and down gives a positive slope.

1. Label: (x_1, y_1) = (-5, 3), (x_2, y_2) = (-1, -9).
2. Numerator: y_2 - y_1 = -9 - 3 = -12. Kim got this right.
3. Denominator: x_2 - x_1 = -1 - (-5) = -1 + 5 = 4. Kim wrote -1 - 5, dropping the negative sign of x_1 = -5. That is the error.
4. Correct slope: m = \frac{-12}{4} = -3

The denominator should be -1 - (-5) = 4, so the slope is -3.

Check: From (-5, 3) to (-1, -9), x increases and y decreases, so the slope must be negative — Kim's positive answer could not be right ✓

Tip: Put each coordinate in parentheses when you substitute: (-1) - (-5). That keeps the negative sign from disappearing.