Line from slope and a point

1. The point (0, -3) has x = 0, so it lies on the y-axis. That makes it the y-intercept: b = -3.
2. The slope is given: m = 4.
3. Substitute into slope-intercept form y = mx + b: y = 4x + (-3) = 4x - 3

Answer: y = 4x - 3

Check: Put x = 0 into y = 4x - 3: 4(0) - 3 = -3 ✓ The line passes through (0, -3), and the number in front of x is 4 ✓

Tip: When the given point has x = 0, you get b for free — no solving needed. The number in front of x is always the slope; the number standing alone (with its sign) is always the y-intercept.

1. Point-slope form (on the formula sheet): y - y_1 = m(x - x_1).
2. Identify the parts: m = 2; the point (3, 5) gives x_1 = 3 and y_1 = 5.
3. Substitute — the y-value goes with y, the x-value goes with x: y - 5 = 2(x - 3)

Answer: y - 5 = 2(x - 3)

Check: Put the point in: left side 5 - 5 = 0; right side 2(3 - 3) = 0 ✓ Both sides match, so (3, 5) is on the line. In slope-intercept form this is y = 2x - 1, and 2(3) - 1 = 5 ✓

Tip: In point-slope form, the point's coordinates appear with minus signs, each next to its own letter: y with y_1, x with x_1.

1. The slope m is the change in y for each 1-unit move to the right. Going down 5 means m = -5.
2. The line crosses the y-axis at (0, 6), so the y-intercept is b = 6.
3. Put both numbers into slope-intercept form y = mx + b: y = -5x + 6

Answer: y = -5x + 6

Check: At x = 0: y = -5(0) + 6 = 6 ✓ (the point (0, 6)). At x = 1: y = -5 + 6 = 1, which is 5 units lower than 6 ✓.

Tip: In y = mx + b the number multiplying x is always the slope and the number standing alone is the y-intercept. "Down" or "falls" means a negative slope.

1. Point-slope form: y - y_1 = m(x - x_1).
2. From the point (-2, 4): x_1 = -2 and y_1 = 4. The slope is m = -3.
3. Substitute: y - 4 = -3\big(x - (-2)\big).
4. Subtracting a negative is adding: x - (-2) = x + 2. y - 4 = -3(x + 2)

Answer: y - 4 = -3(x + 2)

Check: Put in the point (-2, 4): left side 4 - 4 = 0; right side -3(-2 + 2) = -3(0) = 0 ✓.

Tip: In point-slope form the signs inside the equation are the opposite of the signs in the point. The point (-2, 4) shows up as (x + 2) and (y - 4).

1. Start with y = -2x + b.
2. Substitute the point (-1, 4): 4 = -2(-1) + b
3. Multiply: -2 \times (-1) = +2, so 4 = 2 + b.
4. Subtract 2: b = 2.
5. Write the equation: y = -2x + 2.

Another way: point-slope form y - 4 = -2(x + 1); distribute: y - 4 = -2x - 2; add 4: y = -2x + 2 ✓

Answer: y = -2x + 2

Check: -2(-1) + 2 = 2 + 2 = 4 ✓

Tip: Negative × negative = positive. When the point has a negative x-coordinate, wrap it in parentheses before you multiply.

1. The rate is $60 per hour, so the slope is m = 60: C = 60h + b.
2. "A 3-hour job costs $230" gives the point (h, C) = (3, 230).
3. Substitute: 230 = 60(3) + b.
4. Multiply: 230 = 180 + b.
5. Subtract 180: b = 50. The flat fee is $50.
6. Equation: C = 60h + 50.

Answer: C = 60h + 50

Check: 60(3) + 50 = 180 + 50 = 230 ✓ And a 1-hour job would cost 60 + 50 = 110 dollars, which makes sense.

Tip: In a cost problem, the slope is the "per" amount and b is the one-time fee. The total is a point on the line — never the fee itself.

1. The rate is 3^\circ\text{F} per hour, so m = 3.
2. 6 a.m. is 6 hours after midnight: the point is (t_1, T_1) = (6, 48).
3. Point-slope form with T in place of y and t in place of x: T - T_1 = m(t - t_1) T - 48 = 3(t - 6)
4. (Optional) In slope-intercept form: T - 48 = 3t - 18, so T = 3t + 30.

Answer: T - 48 = 3(t - 6)

Check: At t = 6: T - 48 = 3(0) = 0, so T = 48 ✓ At 8 a.m. (t = 8): T = 3(8) + 30 = 54, which is 2 hours × 3° above 48 ✓

Tip: The time goes with t (like x) and the temperature with T (like y). Decide which quantity depends on which before you substitute.

1. Step 1 is correct: slope-intercept form with m = -4.
2. Step 2 is correct: the point gives x = 2, y = -3.
3. Step 3 is correct: -4(2) = -8.
4. Step 4 is wrong. To undo -8, add 8 to both sides: -3 + 8 = b \quad\Rightarrow\quad b = 5 Dana subtracted 8 instead, getting -11.
5. The correct equation is y = -4x + 5.

Answer: Step 4

Check: -4(2) + 5 = -8 + 5 = -3 ✓, but Dana's line gives -4(2) - 11 = -19 \ne -3.

Tip: Always plug the point back into your final equation. A 5-second check catches add/subtract mix-ups like this one.

1. The rate per mile is the slope: m = 2.50. So C = 2.50d + b, where b is the pick-up fee.
2. The 8-mile ride is the point (d, C) = (8, 23.50). Substitute: 23.50 = 2.50(8) + b
3. Multiply: 2.50 \times 8 = 20, so 23.50 = 20 + b.
4. Subtract 20: b = 3.50. The pick-up fee is $3.50.

Answer: C = 2.50d + 3.50

Check: 2.50(8) + 3.50 = 20 + 3.50 = 23.50 ✓.

Tip: The total for one ride ($23.50) is a point on the line, not the starting fee. The fee is what you pay for 0 miles.

1. Point-slope form with m = \tfrac{3}{4} and (8, 1): y - 1 = \tfrac{3}{4}(x - 8)
2. Substitute x = -4: y - 1 = \tfrac{3}{4}(-4 - 8) = \tfrac{3}{4}(-12).
3. Multiply: \tfrac{3}{4} \times (-12) = -9, so y - 1 = -9.
4. Add 1: y = -8.

Answer: -8

Check: Slope-intercept form: b = 1 - \tfrac{3}{4}(8) = 1 - 6 = -5, so y = \tfrac{3}{4}x - 5. At x = -4: -3 - 5 = -8 ✓.

Tip: Moving from x = 8 to x = -4 is a run of -12. Multiply the run by the slope to get the change in y (-9), then add it to the starting y (1).

1. Start with y = -\tfrac{3}{2}x + b.
2. Substitute x = 4, y = -1: -1 = -\tfrac{3}{2}(4) + b
3. Multiply: -\tfrac{3}{2} \times 4 = -6, so -1 = -6 + b.
4. Add 6 to both sides: b = 5.

Answer: y = -\tfrac{3}{2}x + 5

Check: -\tfrac{3}{2}(4) + 5 = -6 + 5 = -1 ✓.

Tip: With a fraction slope, multiply the point's x-value first — here 4 cancels the denominator 2 and gives a whole number. Then undo the subtraction carefully.

1. Point-slope form is y - y_1 = m(x - x_1) with x_1 = -6, y_1 = 5.
2. The correct first step is y - 5 = -\tfrac{2}{3}\big(x - (-6)\big) = -\tfrac{2}{3}(x + 6). Luis wrote (x - 6), so Step 1 is wrong.
3. Steps 2 and 3 follow correctly from his Step 1, but they describe the wrong line.
4. Correct work: y - 5 = -\tfrac{2}{3}x - 4, then y = -\tfrac{2}{3}x + 1.

Answer: Step 1 is the first error; the line should be y = -\tfrac{2}{3}x + 1.

Check: -\tfrac{2}{3}(-6) + 1 = 4 + 1 = 5 ✓. Luis's line gives -\tfrac{2}{3}(-6) + 9 = 13 \ne 5.

Tip: When a later step "looks right," go back to the very first line. An early sign slip makes every following step wrong, even if the algebra is perfect.

1. Start with y = \tfrac{2}{3}x + b.
2. Substitute x = -6 and y = 2: 2 = \tfrac{2}{3}(-6) + b
3. Multiply: \tfrac{2}{3} \times (-6) = \tfrac{-12}{3} = -4. So 2 = -4 + b.
4. Add 4 to both sides: b = 6.
5. Equation: y = \tfrac{2}{3}x + 6.

Answer: y = \tfrac{2}{3}x + 6

Check: \tfrac{2}{3}(-6) + 6 = -4 + 6 = 2 ✓ Also, from (0, 6) go left 3 and down 2 twice → (-6, 2) ✓

Tip: Positive × negative is negative. Then "undo" the -4 by adding 4. Writing each step on your erasable note board prevents the sign slips that GED distractors are built on.

1. Find b: 3 = -4(1) + b \Rightarrow 3 = -4 + b \Rightarrow b = 7. The line is y = -4x + 7.
2. The point (k, -9) is on the line, so substitute x = k, y = -9: -9 = -4k + 7
3. Subtract 7: -16 = -4k.
4. Divide by -4: k = 4.

Answer: k = 4

Check: Slope from (1, 3) to (4, -9): \dfrac{-9 - 3}{4 - 1} = \dfrac{-12}{3} = -4 ✓.

Tip: Another fast route: y must drop by 3 - (-9) = 12. With a slope of -4, that takes a run of 12 \div 4 = 3, so k = 1 + 3 = 4.

1. From the graph, P = (-3, 2).
2. Down 2 for every 3 to the right: m = \dfrac{-2}{3} = -\tfrac{2}{3}.
3. Substitute into y = -\tfrac{2}{3}x + b: 2 = -\tfrac{2}{3}(-3) + b
4. A negative times a negative is positive: -\tfrac{2}{3}(-3) = 2, so 2 = 2 + b and b = 0.

Answer: y = -\tfrac{2}{3}x

Check: From P(-3, 2) go right 3, down 2 → (0, 0) ✓. The line passes through the origin, so b = 0.

Tip: A y-intercept of 0 is allowed — the equation simply has no constant term. Counting rise and run on the grid confirms it instantly.

1. The battery loses 6% per hour, so m = -6: B = -6t + b.
2. The point is (t, B) = (2.5, 71). Substitute: 71 = -6(2.5) + b
3. Multiply: -6 \times 2.5 = -15, so 71 = -15 + b.
4. Add 15: b = 86. The battery started at 86%.

Answer: B = -6t + 86

Check: -6(2.5) + 86 = -15 + 86 = 71 ✓. It makes sense that the start (86%) is higher than 71%, because the battery is draining.

Tip: For a decreasing quantity, the starting value is bigger than any later value. Use that common-sense test to eliminate answers fast.

1. Step 1 is correct: $45 per day is the rate, and 4 days for $215 is the point (4, 215).
2. Step 2 is correct: substitute into C = 45d + b.
3. Step 3 is correct: 215 - 180 = 35, so the cleaning fee is $35 and C = 45d + 35.
4. Step 4 is wrong: Jordan used only 45d and dropped the $35 fee. The correct prediction is C = 45(10) + 35 = 450 + 35 = 485

Answer: Step 4 (the 10-day rental costs $485)

Check: The model reproduces the given data: 45(4) + 35 = 215 ✓.

Tip: After you find b, write the complete equation before you make any prediction. Using only mx is the same as assuming the line goes through the origin.

1. The slope is m = -0.75.
2. Crossing the y-axis 2.5 units below the origin means the y-intercept is the point (0, -2.5), so b = -2.5.
3. Slope-intercept form: y = mx + b: y = -0.75x - 2.5

Answer: y = -0.75x - 2.5

Check: At x = 0: y = -0.75(0) - 2.5 = -2.5 ✓ (2.5 units below the origin). At x = 4: y = -3 - 2.5 = -5.5, a drop of 3 over a run of 4, which is a slope of -0.75 ✓.

Tip: The form y = m(x - h) describes a line through (h, 0) on the x-axis. Don't confuse it with y = mx + b, which shows the y-axis crossing.

1. Find the slope-intercept form: -2 = 3(-1) + b \Rightarrow -2 = -3 + b \Rightarrow b = 1. The line is y = 3x + 1.
2. Rewrite each choice:
   • y + 2 = 3(x + 1): y + 2 = 3x + 3, so y = 3x + 1 ✓ (point-slope form using (-1, -2)).
   • y - 4 = 3(x - 1): y - 4 = 3x - 3, so y = 3x + 1 ✓ (point-slope form using another point on the line, (1, 4)).
   • y = 3x + 1 ✓.
   • y + 2 = 3(x - 1): y + 2 = 3x - 3, so y = 3x - 5 ✗. This uses the point (1, -2), which is not on the line.

Answer: y + 2 = 3(x - 1)

Check: Test (-1, -2) in y + 2 = 3(x - 1): left 0, right 3(-2) = -6 ✗, so this equation misses the given point.

Tip: Point-slope form can use any point on the line, so a correct answer may not show the point you were given. Converting every choice to y = mx + b makes them easy to compare.