Equation of a Line Passing Through Two Points Worksheets - Free Printable

Problem Analysis:

The task involves finding the equation of a line passing through two given points and expressing it in standard form. The standard form of a linear equation is:

\[
Ax + By = C
\]

where \( A \), \( B \), and \( C \) are integers, and \( A \geq 0 \).

#### Part A:
Find the equation of the line passing through the points \((-1, -3)\) and \((2, 1)\).

#### Part B:
Find the equation of the line passing through the points \((4, -5)\) and \((-2, 7)\).

---

Solution:

#### Part A:
1. Calculate the slope (\( m \)):
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the points \((-1, -3)\) and \((2, 1)\):
\[
m = \frac{1 - (-3)}{2 - (-1)} = \frac{1 + 3}{2 + 1} = \frac{4}{3}
\]

2. Use the point-slope form:
The point-slope form of a line is:
\[
y - y_1 = m(x - x_1)
\]
Using the point \((-1, -3)\) and the slope \( m = \frac{4}{3} \):
\[
y - (-3) = \frac{4}{3}(x - (-1))
\]
Simplify:
\[
y + 3 = \frac{4}{3}(x + 1)
\]

3. Eliminate the fraction:
Multiply through by 3 to clear the denominator:
\[
3(y + 3) = 4(x + 1)
\]
Distribute:
\[
3y + 9 = 4x + 4
\]

4. Rearrange into standard form:
Move all terms to one side:
\[
-4x + 3y + 9 - 4 = 0
\]
Simplify:
\[
-4x + 3y + 5 = 0
\]
To ensure \( A \geq 0 \), multiply through by \(-1\):
\[
4x - 3y - 5 = 0
\]

Thus, the equation in standard form is:
\[
\boxed{4x - 3y = 5}
\]

---

#### Part B:
1. Calculate the slope (\( m \)):
Using the points \((4, -5)\) and \((-2, 7)\):
\[
m = \frac{7 - (-5)}{-2 - 4} = \frac{7 + 5}{-2 - 4} = \frac{12}{-6} = -2
\]

2. Use the point-slope form:
Using the point \((4, -5)\) and the slope \( m = -2 \):
\[
y - (-5) = -2(x - 4)
\]
Simplify:
\[
y + 5 = -2(x - 4)
\]

3. Eliminate the parentheses:
Distribute the \(-2\):
\[
y + 5 = -2x + 8
\]

4. Rearrange into standard form:
Move all terms to one side:
\[
2x + y + 5 - 8 = 0
\]
Simplify:
\[
2x + y - 3 = 0
\]

Thus, the equation in standard form is:
\[
\boxed{2x + y = 3}
\]

---

Final Answers:

- Part A: \(\boxed{4x - 3y = 5}\)
- Part B: \(\boxed{2x + y = 3}\)