Answer :
Let's solve the problem step-by-step. We need to find the equation of the line that passes through the points (5, -7) and (-1, 5). The equation of a line in slope-intercept form is given by:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
### Step 1: Calculate the Slope [tex]\( m \)[/tex]
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Let's substitute the given points [tex]\((5, -7)\)[/tex] and [tex]\((-1, 5)\)[/tex]:
[tex]\[ m = \frac{5 - (-7)}{-1 - 5} \][/tex]
[tex]\[ m = \frac{5 + 7}{-1 - 5} \][/tex]
[tex]\[ m = \frac{12}{-6} \][/tex]
[tex]\[ m = -2.0 \][/tex]
### Step 2: Calculate the Y-Intercept [tex]\( b \)[/tex]
The y-intercept [tex]\( b \)[/tex] can be found using the slope-intercept form [tex]\( y = mx + b \)[/tex] by substituting one of the points and the slope. We'll use the point [tex]\((5, -7)\)[/tex]:
[tex]\[ -7 = (-2.0)(5) + b \][/tex]
[tex]\[ -7 = -10 + b \][/tex]
[tex]\[ b = -7 + 10 \][/tex]
[tex]\[ b = 3.0 \][/tex]
### Step 3: Write the Equation of the Line
Now that we have the slope [tex]\( m = -2.0 \)[/tex] and the y-intercept [tex]\( b = 3.0 \)[/tex], we can write the equation of the line:
[tex]\[ y = -2.0x + 3.0 \][/tex]
### Summary
The equation of the line passing through the points (5, -7) and (-1, 5) is:
[tex]\[ y = -2.0x + 3.0 \][/tex]
### Graphing the Line
1. Plot the Points: Start by plotting the given points (5, -7) and (-1, 5) on the coordinate plane.
2. Draw the Line: Use a ruler to draw a straight line through these two points. This line represents the equation [tex]\( y = -2.0x + 3.0 \)[/tex].
To assist with graphing:
- The y-intercept [tex]\( b = 3.0 \)[/tex] means the line crosses the y-axis at (0, 3).
- The slope [tex]\( m = -2.0 \)[/tex] indicates that for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 2 units.
This will help ensure the line is drawn accurately on the graph.
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
### Step 1: Calculate the Slope [tex]\( m \)[/tex]
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Let's substitute the given points [tex]\((5, -7)\)[/tex] and [tex]\((-1, 5)\)[/tex]:
[tex]\[ m = \frac{5 - (-7)}{-1 - 5} \][/tex]
[tex]\[ m = \frac{5 + 7}{-1 - 5} \][/tex]
[tex]\[ m = \frac{12}{-6} \][/tex]
[tex]\[ m = -2.0 \][/tex]
### Step 2: Calculate the Y-Intercept [tex]\( b \)[/tex]
The y-intercept [tex]\( b \)[/tex] can be found using the slope-intercept form [tex]\( y = mx + b \)[/tex] by substituting one of the points and the slope. We'll use the point [tex]\((5, -7)\)[/tex]:
[tex]\[ -7 = (-2.0)(5) + b \][/tex]
[tex]\[ -7 = -10 + b \][/tex]
[tex]\[ b = -7 + 10 \][/tex]
[tex]\[ b = 3.0 \][/tex]
### Step 3: Write the Equation of the Line
Now that we have the slope [tex]\( m = -2.0 \)[/tex] and the y-intercept [tex]\( b = 3.0 \)[/tex], we can write the equation of the line:
[tex]\[ y = -2.0x + 3.0 \][/tex]
### Summary
The equation of the line passing through the points (5, -7) and (-1, 5) is:
[tex]\[ y = -2.0x + 3.0 \][/tex]
### Graphing the Line
1. Plot the Points: Start by plotting the given points (5, -7) and (-1, 5) on the coordinate plane.
2. Draw the Line: Use a ruler to draw a straight line through these two points. This line represents the equation [tex]\( y = -2.0x + 3.0 \)[/tex].
To assist with graphing:
- The y-intercept [tex]\( b = 3.0 \)[/tex] means the line crosses the y-axis at (0, 3).
- The slope [tex]\( m = -2.0 \)[/tex] indicates that for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 2 units.
This will help ensure the line is drawn accurately on the graph.
