Question

for each equation, state the slope and the y-intercept from the equation. the y-intercept is to be stated as an ordered pair (not one number). then graph the line. 1. $y = \frac{2}{3}x + 1$ 2. $y = \frac{5}{8}x - 3$ 3. $y = 3x + 5$ 4. $y = -\frac{3}{5}x + 4$ 5. $y = -\frac{1}{4}x - 2$ 6. $y = -5x$

Explanation:

Step1: Recall Slope-Intercept Form

The slope - intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept (the value of \(y\) when \(x = 0\)). The \(y\) - intercept as an ordered pair is \((0,b)\).

Step2: Analyze Equation 1 (\(y=\frac{2}{3}x + 1\))

• For the equation \(y=\frac{2}{3}x + 1\), comparing with \(y=mx + b\):
• The slope \(m=\frac{2}{3}\).
• The \(y\) - intercept \(b = 1\), so the \(y\) - intercept as an ordered pair is \((0,1)\).
• To graph the line: Plot the \(y\) - intercept \((0,1)\). Then, use the slope \(\frac{2}{3}\) (rise = 2, run = 3). From \((0,1)\), move up 2 units and right 3 units to get another point \((3,3)\), and draw a line through these points.

Step3: Analyze Equation 2 (\(y=\frac{5}{8}x-3\))

• For the equation \(y=\frac{5}{8}x - 3\), comparing with \(y = mx + b\):
• The slope \(m=\frac{5}{8}\).
• The \(y\) - intercept \(b=-3\), so the \(y\) - intercept as an ordered pair is \((0, - 3)\).
• To graph the line: Plot the \(y\) - intercept \((0,-3)\). Then, use the slope \(\frac{5}{8}\) (rise = 5, run = 8). From \((0,-3)\), move up 5 units and right 8 units to get another point \((8,2)\), and draw a line through these points.

Step4: Analyze Equation 3 (\(y = 3x+5\))

• For the equation \(y = 3x + 5\), comparing with \(y=mx + b\):
• The slope \(m = 3=\frac{3}{1}\).
• The \(y\) - intercept \(b = 5\), so the \(y\) - intercept as an ordered pair is \((0,5)\).
• To graph the line: Plot the \(y\) - intercept \((0,5)\). Then, use the slope \(\frac{3}{1}\) (rise = 3, run = 1). From \((0,5)\), move up 3 units and right 1 unit to get another point \((1,8)\), and draw a line through these points.

Step5: Analyze Equation 4 (\(y=-\frac{3}{5}x + 4\))

• For the equation \(y=-\frac{3}{5}x + 4\), comparing with \(y=mx + b\):
• The slope \(m=-\frac{3}{5}\).
• The \(y\) - intercept \(b = 4\), so the \(y\) - intercept as an ordered pair is \((0,4)\).
• To graph the line: Plot the \(y\) - intercept \((0,4)\). Then, use the slope \(-\frac{3}{5}\) (rise=- 3, run = 5 or rise = 3, run=-5). From \((0,4)\), move down 3 units and right 5 units to get another point \((5,1)\), and draw a line through these points.

Step6: Analyze Equation 5 (\(y=-\frac{1}{4}x-2\))

• For the equation \(y=-\frac{1}{4}x - 2\), comparing with \(y=mx + b\):
• The slope \(m=-\frac{1}{4}\).
• The \(y\) - intercept \(b=-2\), so the \(y\) - intercept as an ordered pair is \((0,-2)\).
• To graph the line: Plot the \(y\) - intercept \((0,-2)\). Then, use the slope \(-\frac{1}{4}\) (rise=-1, run = 4 or rise = 1, run=-4). From \((0,-2)\), move down 1 unit and right 4 units to get another point \((4,-3)\), and draw a line through these points.

Step7: Analyze Equation 6 (\(y=-5x\))

• For the equation \(y=-5x\), we can rewrite it as \(y=-5x+0\). Comparing with \(y = mx + b\):
• The slope \(m=-5=\frac{-5}{1}\).
• The \(y\) - intercept \(b = 0\), so the \(y\) - intercept as an ordered pair is \((0,0)\) (the origin).
• To graph the line: Plot the \(y\) - intercept \((0,0)\). Then, use the slope \(-5\) (rise=-5, run = 1 or rise = 5, run=-1). From \((0,0)\), move down 5 units and right 1 unit to get the point \((1,-5)\), and draw a line through these points.

Answer:

s (for each equation):

1. Equation \(y=\frac{2}{3}x + 1\): Slope \(=\frac{2}{3}\), \(y\) - intercept \(=(0,1)\)
2. Equation \(y=\frac{5}{8}x-3\): Slope \(=\frac{5}{8}\), \(y\) - intercept \(=(0, - 3)\)
3. Equation \(y = 3x+5\): Slope \(=3\), \(y\) - intercept \(=(0,5)\)
4. Equation \(y=-\frac{3}{5}x + 4\): Slope \(=-\frac{3}{5}\), \(y\) - intercept \(=(0,4)\)
5. Equation \(y=-\frac{1}{4}x-2\): Slope \(=-\frac{1}{4}\), \(y\) - intercept \(=(0,-2)\)
6. Equation \(y=-5x\): Slope \(=-5\), \(y\) - intercept \(=(0,0)\)