Question

write the standard form of the equation of the line through the given point with the given slope.

1. through: (4, -2), slope = -1
2. through: (-2, 4), slope = -\frac{1}{7}

write the standard form of the equation of the line through the given points.

1. through: (-3, 2) and (0, -1)
2. through: (0, 4) and (-1, -1)

write the standard form of the equation of the line described.

1. through: (2, 0), parallel to y = \frac{2}{3}x
2. through: (-2, 4), parallel to y = -\frac{3}{2}x + 3
3. through: (2, 4), perp. to y = -\frac{2}{7}x - 5
4. through: (5, 0), perp. to y = -x + 5

Explanation:

Let's solve problem 15: Write the standard form of the line through \((4, -2)\) with slope \(=-1\).

Step 1: Use point - slope form

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(4,-2)\) and \(m = - 1\).
Substitute the values into the formula: \(y-(-2)=-1(x - 4)\)
Simplify the left - hand side and the right - hand side: \(y + 2=-x + 4\)

Step 2: Convert to standard form

The standard form of a line is \(Ax+By = C\), where \(A\), \(B\), and \(C\) are integers and \(A\geq0\).
Add \(x\) to both sides of the equation \(y + 2=-x + 4\): \(x+y+2 = 4\)
Subtract 2 from both sides: \(x + y=2\)

Answer:

For problem 15, the standard form of the line is \(x + y=2\)