Graphing and writing linear equations in point-slope form worksheet with four coordinate grids.

Let’s solve each problem step by step. We are given linear equations in point-slope form and asked to graph them and write the equation in slope-intercept form (y = mx + b).

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Problem 1: Graph y - 2 = ½(x - 4)

Step 1: Identify the point and slope from point-slope form.
Point-slope form is: y - y₁ = m(x - x₁)
So here, point (x₁, y₁) = (4, 2), and slope m = ½.

Step 2: Plot the point (4, 2) on the grid.

Step 3: Use the slope ½ to find another point.
Slope = rise/run = 1/2 → go up 1 unit, right 2 units from (4, 2) → new point: (6, 3)
Or go down 1, left 2 → (2, 1)

Step 4: Draw a line through these points.

Step 5: Convert to slope-intercept form (y = mx + b)
Start with: y - 2 = ½(x - 4)
Distribute: y - 2 = ½x - 2
Add 2 to both sides: y = ½x - 2 + 2 → y = ½x

✔ So for Problem 1:
- Slope: ½
- Y-intercept: 0
- Equation: y = ½x

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Problem 2: Graph y + 1 = -(x + 2)

Step 1: Rewrite to match point-slope form: y - (-1) = -1(x - (-2))
So point (x₁, y₁) = (-2, -1), slope m = -1

Step 2: Plot point (-2, -1)

Step 3: Use slope -1 = -1/1 → go down 1, right 1 → from (-2, -1) → (-1, -2)
Or up 1, left 1 → (-3, 0)

Step 4: Draw line through those points.

Step 5: Convert to slope-intercept form:
y + 1 = -1(x + 2)
→ y + 1 = -x - 2
Subtract 1: y = -x - 3

✔ So for Problem 2:
- Slope: -1
- Y-intercept: -3
- Equation: y = -x - 3

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Problem 3: Graph y + 3 = -⅓(x - 6)

Step 1: Point-slope form: y - (-3) = -⅓(x - 6)
Point: (6, -3), slope: -⅓

Step 2: Plot (6, -3)

Step 3: Slope -⅓ → down 1, right 3 → (9, -4)
Or up 1, left 3 → (3, -2)

Step 4: Draw line.

Step 5: Convert to slope-intercept:
y + 3 = -⅓(x - 6)
→ y + 3 = -⅓x + 2 [because -⅓ * -6 = +2]
Subtract 3: y = -⅓x + 2 - 3 → y = -⅓x - 1

✔ So for Problem 3:
- Slope: -
- Y-intercept: -1
- Equation: y = -⅓x - 1

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Problem 4: Graph y - 4 = ¾(x + 1)

Step 1: Rewrite as y - 4 = ¾(x - (-1))
Point: (-1, 4), slope: ¾

Step 2: Plot (-1, 4)

Step 3: Slope ¾ → up 3, right 4 → from (-1, 4) → (3, 7)
Or down 3, left 4 → (-5, 1)

Step 4: Draw line.

Step 5: Convert to slope-intercept:
y - 4 = ¾(x + 1)
→ y - 4 = ¾x + ¾
Add 4: y = ¾x + ¾ + 4 → y = ¾x + 4.75 or better as fraction: 4 = 16/4 → ¾ + 16/4 = 19/4? Wait — let's do it properly:

¾ + 4 = ¾ + 16/4 = 19/4? No — 4 is 16/4? Actually, 4 = 16/4 is wrong — 4 = 16/4? 16÷4=4 → yes, but we’re adding ¾ + 4.

Better: 4 = 16/4? No — 4 = 16/4 is correct numerically, but let’s use common denominator:

¾ + 4 = ¾ + 16/4? Wait — 4 = 16/4? That’s 4.0, yes. But actually:

We have: y = ¾x + ¾ + 4

Convert 4 to fourths: 4 = 16/4? No — 4 = 16/4 is incorrect because 16/4 = 4, yes, but that’s not helping.

Actually: ¾ + 4 = ¾ + 4/1 = ¾ + 16/4? No — 4/1 = 16/4? 4/1 = 16/4 → 4 = 4, yes, but 16/4 is 4, so:

¾ + 4 = ¾ + 16/4? That would be if we wrote 4 as 16/4, which is fine, but 16/4 is 4, so:

¾ + 16/4 = (3 + 16)/4 = 19/4? But 19/4 is 4.75 — that’s correct.

But let me recalculate:

y - 4 = ¾(x + 1)
→ y - 4 = ¾x + ¾
→ y = ¾x + ¾ + 4
Now, 4 = 16/4? Yes, because 16 ÷ 4 = 4. So:

¾ + 16/4 = (3 + 16)/4 = 19/4 → y = ¾x + 19/4

But 19/4 is 4.75 — that’s acceptable, but maybe leave as improper fraction or mixed number? The problem doesn’t specify, but usually slope-intercept uses fractions or decimals. Since slope is fraction, keep as fraction.

Alternatively, check calculation again:

¾ + 4 = ¾ + 4.0 = 4.75 = 19/4 — yes.

But wait — is there a mistake? Let me plug in x = -1 into final equation to verify:

Original: when x = -1, y should be 4.

From y = ¾x + 19/4 → at x = -1: y = ¾(-1) + 19/4 = -¾ + 19/4 = 16/4 = 4 → correct.

At x = 3: y = ¾(3) + 19/4 = 9/4 + 19/4 = 28/4 = 7 → matches our earlier point (3,7). Good.

✔ So for Problem 4:
- Slope: ¾
- Y-intercept: 19/4 (or 4.75)
- Equation: y = ¾x + 19/4

But perhaps simplify 19/4 as mixed number? 4 and 3/4? But in algebra, improper fraction is fine.

However, let me double-check the addition:

¾ + 4 = ?

Write 4 as 16/4? No — 4 = 16/4 is true, but 16/4 is 4, so ¾ + 16/4 = 19/4 — yes.

But actually, 4 = 16/4 is redundant — just think: 4 = 4.0, and ¾ = 0.75, sum 4.75 = 19/4.

Yes.

Alternatively, some might prefer to write as y = (3/4)x + 19/4.

All good.

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Final Answers Summary:

Problem 1:
- Slope: 1/2
- Y-intercept: 0
- Equation: y = (1/2)x

Problem 2:
- Slope: -1
- Y-intercept: -3
- Equation: y = -x - 3

Problem 3:
- Slope: -1/3
- Y-intercept: -1
- Equation: y = (-1/3)x - 1

Problem 4:
- Slope: 3/4
- Y-intercept: 19/4
- Equation: y = (3/4)x + 19/4

Note: For Problem 4, y-intercept can also be written as 4.75, but since other answers use fractions, we’ll keep 19/4.

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Final Answer:

1. Slope: \frac{1}{2}, Y-intercept: 0, Equation: y = \frac{1}{2}x
2. Slope: -1, Y-intercept: -3, Equation: y = -x - 3
3. Slope: -\frac{1}{3}, Y-intercept: -1, Equation: y = -\frac{1}{3}x - 1
4. Slope: \frac{3}{4}, Y-intercept: \frac{19}{4}, Equation: y = \frac{3}{4}x + \frac{19}{4}