Question

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Explanation:

To solve for the equations of the lines in each graph, we use the slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept. The slope \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}\)

Problem 5

Step 1: Identify two points on the line

From the graph, we can see that the line passes through \((6,6)\) and \((9,1)\)

Step 2: Calculate the slope (\(m\))

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \(x_1 = 6,y_1 = 6,x_2=9,y_2 = 1\)
\(m=\frac{1 - 6}{9 - 6}=\frac{- 5}{3}\)

Step 3: Find the \(y\) - intercept (\(b\))

We use the point - slope form \(y - y_1=m(x - x_1)\). Let's use the point \((6,6)\)
\(y-6=\frac{-5}{3}(x - 6)\)
\(y-6=\frac{-5}{3}x + 10\)
\(y=\frac{-5}{3}x+10 + 6\)
\(y=\frac{-5}{3}x + 16\) (or we can also use the slope - intercept form directly. Substitute \(x = 6,y = 6\) and \(m=\frac{-5}{3}\) into \(y=mx + b\)
\(6=\frac{-5}{3}(6)+b\)
\(6=- 10 + b\)
\(b=16\))

Problem 6

Step 1: Identify two points on the line

The line is a vertical line. For a vertical line, the equation is of the form \(x = k\), where \(k\) is the \(x\) - coordinate of any point on the line. From the graph, we can see that for all points on the line, \(x = 3\)

Problem 7

Step 1: Identify two points on the line

The line passes through \((0,4)\) and \((8,7)\)

Step 2: Calculate the slope (\(m\))

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \(x_1 = 0,y_1 = 4,x_2 = 8,y_2=7\)
\(m=\frac{7 - 4}{8 - 0}=\frac{3}{8}\)

Step 3: Find the \(y\) - intercept (\(b\))

Since the line passes through \((0,4)\), when \(x = 0,y = 4\). In the slope - intercept form \(y=mx + b\), when \(x = 0,y=b\). So \(b = 4\)
The equation of the line is \(y=\frac{3}{8}x + 4\)

Problem 8

Step 1: Identify two points on the line

The line passes through \((4,9)\) and \((6,7)\)

Step 2: Calculate the slope (\(m\))

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \(x_1 = 4,y_1 = 9,x_2 = 6,y_2=7\)
\(m=\frac{7 - 9}{6 - 4}=\frac{-2}{2}=-1\)

Step 3: Find the \(y\) - intercept (\(b\))

Substitute \(x = 4,y = 9\) and \(m=-1\) into \(y=mx + b\)
\(9=-1\times4 + b\)
\(9=-4 + b\)
\(b=13\)
The equation of the line is \(y=-x + 13\)

Problem 9

Step 1: Identify two points on the line

The line passes through \((5,2)\) and \((9,9)\)

Step 2: Calculate the slope (\(m\))

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \(x_1 = 5,y_1 = 2,x_2 = 9,y_2=9\)
\(m=\frac{9 - 2}{9 - 5}=\frac{7}{4}\)

Step 3: Find the \(y\) - intercept (\(b\))

Substitute \(x = 5,y = 2\) and \(m=\frac{7}{4}\) into \(y=mx + b\)
\(2=\frac{7}{4}(5)+b\)
\(2=\frac{35}{4}+b\)
\(b=2-\frac{35}{4}=\frac{8 - 35}{4}=\frac{-27}{4}\)
The equation of the line is \(y=\frac{7}{4}x-\frac{27}{4}\)

Final Answers

1. \(\boldsymbol{y =-\frac{5}{3}x + 16}\)

1. \(\boldsymbol{x = 3}\)

1. \(\boldsymbol{y=\frac{3}{8}x + 4}\)

1. \(\boldsymbol{y=-x + 13}\)

1. \(\boldsymbol{y=\frac{7}{4}x-\frac{27}{4}}\)

Answer:

To solve for the equations of the lines in each graph, we use the slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept. The slope \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}\)

Problem 5

Step 1: Identify two points on the line

From the graph, we can see that the line passes through \((6,6)\) and \((9,1)\)

Step 2: Calculate the slope (\(m\))

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \(x_1 = 6,y_1 = 6,x_2=9,y_2 = 1\)
\(m=\frac{1 - 6}{9 - 6}=\frac{- 5}{3}\)

Step 3: Find the \(y\) - intercept (\(b\))

We use the point - slope form \(y - y_1=m(x - x_1)\). Let's use the point \((6,6)\)
\(y-6=\frac{-5}{3}(x - 6)\)
\(y-6=\frac{-5}{3}x + 10\)
\(y=\frac{-5}{3}x+10 + 6\)
\(y=\frac{-5}{3}x + 16\) (or we can also use the slope - intercept form directly. Substitute \(x = 6,y = 6\) and \(m=\frac{-5}{3}\) into \(y=mx + b\)
\(6=\frac{-5}{3}(6)+b\)
\(6=- 10 + b\)
\(b=16\))

Problem 6

Step 1: Identify two points on the line

The line is a vertical line. For a vertical line, the equation is of the form \(x = k\), where \(k\) is the \(x\) - coordinate of any point on the line. From the graph, we can see that for all points on the line, \(x = 3\)

Problem 7

Step 1: Identify two points on the line

The line passes through \((0,4)\) and \((8,7)\)

Step 2: Calculate the slope (\(m\))

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \(x_1 = 0,y_1 = 4,x_2 = 8,y_2=7\)
\(m=\frac{7 - 4}{8 - 0}=\frac{3}{8}\)

Step 3: Find the \(y\) - intercept (\(b\))

Since the line passes through \((0,4)\), when \(x = 0,y = 4\). In the slope - intercept form \(y=mx + b\), when \(x = 0,y=b\). So \(b = 4\)
The equation of the line is \(y=\frac{3}{8}x + 4\)

Problem 8

Step 1: Identify two points on the line

The line passes through \((4,9)\) and \((6,7)\)

Step 2: Calculate the slope (\(m\))

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \(x_1 = 4,y_1 = 9,x_2 = 6,y_2=7\)
\(m=\frac{7 - 9}{6 - 4}=\frac{-2}{2}=-1\)

Step 3: Find the \(y\) - intercept (\(b\))

Substitute \(x = 4,y = 9\) and \(m=-1\) into \(y=mx + b\)
\(9=-1\times4 + b\)
\(9=-4 + b\)
\(b=13\)
The equation of the line is \(y=-x + 13\)

Problem 9

Step 1: Identify two points on the line

The line passes through \((5,2)\) and \((9,9)\)

Step 2: Calculate the slope (\(m\))

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \(x_1 = 5,y_1 = 2,x_2 = 9,y_2=9\)
\(m=\frac{9 - 2}{9 - 5}=\frac{7}{4}\)

Step 3: Find the \(y\) - intercept (\(b\))

Substitute \(x = 5,y = 2\) and \(m=\frac{7}{4}\) into \(y=mx + b\)
\(2=\frac{7}{4}(5)+b\)
\(2=\frac{35}{4}+b\)
\(b=2-\frac{35}{4}=\frac{8 - 35}{4}=\frac{-27}{4}\)
The equation of the line is \(y=\frac{7}{4}x-\frac{27}{4}\)

Final Answers

1. \(\boldsymbol{y =-\frac{5}{3}x + 16}\)

1. \(\boldsymbol{x = 3}\)

1. \(\boldsymbol{y=\frac{3}{8}x + 4}\)

1. \(\boldsymbol{y=-x + 13}\)

1. \(\boldsymbol{y=\frac{7}{4}x-\frac{27}{4}}\)