A line is shown on the coordinate plane.
On coordinate plane, points (2, 1), (1, 5) and (0, 9) are connected through a line.
Drag a number to each box to represent the equation of the line.
The equation of a line can be represented in the form y = mx + b, where m is the slope and b is the y-intercept.
To find the slope (m) of the line, we can use the formula:
m = (change in y) / (change in x)
First, find the change in y: (y2 - y1) = (5 - 1) = 4
Then, find the change in x: (x2 - x1) = (1 - 2) = -1
Now, calculate the slope:
m = 4 / -1 = -4
So far, the equation is y = -4x + b.
To find the y-intercept (b), we can substitute one of the given points into the equation. Let's use the point (2, 1):
1 = -4(2) + b
1 = -8 + b
b = 1 + 8
b = 9
Now we have the complete equation:
y = -4x + 9
The graphs of two lines are shown.
Graph 1 titled 'Line A' has the intersecting points at (0, 7) and (2, 1). Graph 2 titled 'Line B' has the intersecting points at (0, 0) and (3, 7).
Drag the correct equation for each line to the appropriate box.
For Line A:
To find the equation of Line A, we can use the formula for finding the equation of a line:
y = mx + b,
where m is the slope and b is the y-intercept.
To find the slope (m) of Line A, we can use the formula:
m = (change in y) / (change in x).
Using the points (0, 7) and (2, 1), we have:
change in y = 1 - 7 = -6,
change in x = 2 - 0 = 2.
Thus, the slope of Line A is:
m = -6 / 2 = -3.
Substituting the slope and a point (0, 7) into the equation y = mx + b, we can solve for the y-intercept (b):
7 = -3(0) + b,
7 = b.
Therefore, the equation of Line A is:
y = -3x + 7.
For Line B:
Using the points (0, 0) and (3, 7), we can find the slope (m) of Line B:
change in y = 7 - 0 = 7,
change in x = 3 - 0 = 3.
Thus, the slope of Line B is:
m = 7 / 3.
Substituting the slope and a point (0, 0) into the equation y = mx + b, we can solve for the y-intercept (b):
0 = (7/3)(0) + b,
0 = b.
Therefore, the equation of Line B is:
y = (7/3)x + 0.
Simplifying the equation, Line B can also be written as:
y = (7/3)x.
In summary, the equations for Line A and Line B are:
Line A: y = -3x + 7,
Line B: y = (7/3)x.
A salesman makes both a base salary and also a commission, which is a percentage of what he sells. Each month, if his sales total s dollars, he makes a total of 2,000 + 0.1s dollars. What does 2,000 represent?
A.
his monthly base salary in dollars
B.
the amount of his monthly sales in dollars
C.
his total monthly pay in dollars
D.
his monthly commission in dollars
The correct answer is D. his monthly commission in dollars.
In the given situation, the salesman's total monthly pay is represented by 2,000 + 0.1s. The term 2,000 represents the base salary, which is a fixed amount that he receives regardless of his sales. The term 0.1s represents his commission, which is a percentage (0.1 or 10% in this case) of his sales. So, 2,000 represents his monthly commission in dollars.
To determine the equation of the line passing through the points (2, 1), (1, 5), and (0, 9), we can use the slope-intercept form of a linear equation, which is:
y = mx + b
where m represents the slope of the line, and b represents the y-intercept.
Step 1: Calculate the slope (m)
The slope (m) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Using the points (2, 1) and (1, 5), we have:
m = (5 - 1) / (1 - 2)
m = 4 / -1
m = -4
So, the slope (m) of the line is -4.
Step 2: Calculate the y-intercept (b)
To find the y-intercept (b), we can use one of the given points. Let's use the point (0, 9), and substitute the coordinates (x, y) into the equation:
y = mx + b
9 = -4 * 0 + b
9 = b
So, the y-intercept (b) of the line is 9.
Step 3: Write the equation of the line
Now that we have the values of m and b, we can substitute them into the slope-intercept form of the equation:
y = mx + b
Substituting -4 for m and 9 for b:
y = -4x + 9
Therefore, the equation of the line passing through the given points (2, 1), (1, 5), and (0, 9) is:
y = -4x + 9
To find the equation of a line connecting three given points, we can use the point-slope form of a linear equation.
The point-slope form of a linear equation is given by:
y - y₁ = m(x - x₁)
where (x₁, y₁) is a point on the line, and m is the slope of the line.
Let's calculate the slope (m) using the two given points (2, 1) and (1, 5):
m = (y₂ - y₁) / (x₂ - x₁)
= (5 - 1) / (1 - 2)
= 4 / (-1)
= -4
Now we have the slope (m = -4) and a point on the line (2, 1). We can substitute these values into the point-slope form to get the equation:
y - 1 = -4(x - 2)
Now, let's simplify this equation:
y - 1 = -4x + 8
To further simplify it, move -4x to the left-hand side:
4x + y - 1 = 8
Finally, add 1 to both sides:
4x + y = 9
Therefore, the equation of the line connecting the three given points is 4x + y = 9.