Question

-3x + y = -2 multiplying (-1)
3x - y = 2 standard form
time to practice: write the standard form equation of each line graphed below.
1)
2)
3)
4)
5)
6)

Answer:

To solve these problems, we'll use the standard form of a linear equation, which is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) is non - negative. We'll find two points on each line, calculate the slope (\(m=\frac{y_2 - y_1}{x_2 - x_1}\)), then use the point - slope form \(y - y_1=m(x - x_1)\) and convert it to standard form.

Problem 1

1. Find two points: Let's assume the two points on the line are \((0,3)\) and \((4,2)\).
2. Calculate the slope:

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Substituting \(x_1 = 0,y_1 = 3,x_2 = 4,y_2 = 2\) into the formula, we get \(m=\frac{2 - 3}{4 - 0}=\frac{-1}{4}=-\frac{1}{4}\).

1. Use point - slope form: Using the point \((0,3)\) and the slope \(m =-\frac{1}{4}\) in the point - slope form \(y - y_1=m(x - x_1)\), we have \(y - 3=-\frac{1}{4}(x - 0)\), which simplifies to \(y-3 =-\frac{1}{4}x\).
2. Convert to standard form:

Multiply through by 4 to clear the fraction: \(4y-12=-x\). Then, add \(x\) to both sides and add 12 to both sides to get \(x + 4y=12\).

Problem 2

1. Find two points: Let the two points be \((-2,-1)\) and \((2,1)\).
2. Calculate the slope:

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), with \(x_1=-2,y_1=-1,x_2 = 2,y_2 = 1\), we get \(m=\frac{1-(-1)}{2-(-2)}=\frac{2}{4}=\frac{1}{2}\).

1. Use point - slope form: Using the point \((-2,-1)\) and \(m=\frac{1}{2}\) in the point - slope form \(y - y_1=m(x - x_1)\), we have \(y+1=\frac{1}{2}(x + 2)\).

Expanding the right - hand side gives \(y + 1=\frac{1}{2}x+1\).
Subtract \(y\) from both sides and subtract 1 from both sides: \(0=\frac{1}{2}x - y\).
Multiply through by 2 to get \(x-2y = 0\).

Problem 3

1. Find two points: The two points are \((0,0)\) and \((2,2)\).
2. Calculate the slope:

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), with \(x_1 = 0,y_1 = 0,x_2 = 2,y_2 = 2\), we get \(m=\frac{2 - 0}{2 - 0}=1\).

1. Use point - slope form: Using the point \((0,0)\) and \(m = 1\) in the point - slope form \(y - y_1=m(x - x_1)\), we have \(y-0=1\times(x - 0)\), which simplifies to \(y=x\).
2. Convert to standard form: Subtract \(y\) from both sides to get \(x - y=0\).

Problem 4

1. Find two points: Let the two points be \((-1,2)\) and \((0,0)\).
2. Calculate the slope:

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), with \(x_1=-1,y_1 = 2,x_2 = 0,y_2 = 0\), we get \(m=\frac{0 - 2}{0-(-1)}=\frac{-2}{1}=-2\).

1. Use point - slope form: Using the point \((0,0)\) and \(m=-2\) in the point - slope form \(y - y_1=m(x - x_1)\), we have \(y-0=-2(x - 0)\), which simplifies to \(y=-2x\).
2. Convert to standard form: Add \(2x\) to both sides to get \(2x + y=0\).

Problem 5

1. Find two points: Let the two points be \((0,3)\) and \((2,2)\).
2. Calculate the slope:

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), with \(x_1 = 0,y_1 = 3,x_2 = 2,y_2 = 2\), we get \(m=\frac{2 - 3}{2 - 0}=\frac{-1}{2}=-\frac{1}{2}\).

1. Use point - slope form: Using the point \((0,3)\) and \(m =-\frac{1}{2}\) in the point - slope form \(y - y_1=m(x - x_1)\), we have \(y - 3=-\frac{1}{2}(x - 0)\), which simplifies to \(y-3=-\frac{1}{2}x\).
2. Convert to standard form:

Multiply through by 2 to clear the fraction: \(2y-6=-x\). Then add \(x\) to both sides and add 6 to both sides to get \(x + 2y=6\).

Problem 6

1. Find two points: Let the two points be \((-4,2)\) and \((4,-1)\).
2. Calculate the slope:

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), with \(x_1=-4,y_1 = 2,x_2 = 4,y_2=-1\), we get \(m=\frac{-1 - 2}{4-(-4)}=\frac{-3}{8}=-\frac{3}{8}\).

1. Use point - slope form: Using the point \((-4,2)\) and \(m =-\frac{3}{8}\) in the point - slope form \(y - y_1=m(x - x_1)\), we have \(y - 2=-\frac{3}{8}(x + 4)\).

Expanding the right - hand side: \(y - 2=-\frac{3}{8}x-\frac{3}{2}\).
Add 2 to both sides: \(y=-\frac{3}{8}x-\frac{3}{2}+2=-\frac{3}{8}x+\frac{1}{2}\).

1. Convert to standard form:

Multiply through by 8 to clear the fraction: \(8y=-3x + 4\). Then add \(3x\) to both sides to get \(3x+8y = 4\).

Final Answers

1. \(x + 4y=12\)
2. \(x-2y = 0\)
3. \(x - y=0\)
4. \(2x + y=0\)
5. \(x + 2y=6\)
6. \(3x+8y = 4\)