Find the intersection of the line through (0, 1) and (4.4, 2) and the line through (1.9, 3) and (5.3, 0). (Round your answers to the nearest tenth.)
(x, y) =
(0 , 1) , (4.4 , 2).(1.9 , 3)(5.3 , 0).
m1 = (2 - 1) / (4.4 - 0)
= 1 / 4.4 = 0.227,
y = mx + b,
1 = 0.227*0 + b,
b = 1.
Eq1. y = 0.227x + 1.
(1.9 , 3) , (5.3 , 0),
m2 = (0 - 3) / (5,3 - 1.9),
= -3 / 3.4 = -0.88,
y = mx + b,
3 = -0.88*1.9 + b,
3 = -1.68 + b,
3 + 1.68 = b,
b = 4.68.
Eq2. y = -0.88x + 4.68.
Substitute Eq1 for y in Eq2:
0.227x + 1 = -0.88x + 4.68,
0.227x + 0.88x = 4.68 - 1,
1.107x = 3.68,
x = 3.68 /1.107 = 3.32.
Substitute 3.32 for x in Eq1:
y = 0.227*3.32 + 1,
= 0.755 + 1,
= 1.76.
Solution set: (x , y) = (3.3 , 1.8) =
Point where the 2 lines intersect.
To find the intersection of two lines, we can use the formula for the equation of a line in point-slope form:
(y - y1) = m(x - x1)
Where (x1, y1) is a point on the line and m is the slope of the line.
First, let's find the slope of the first line:
m1 = (y2 - y1) / (x2 - x1)
= (2 - 1) / (4.4 - 0)
= 1 / 4.4
So, the equation of the first line can be written as:
(y - 1) = (1/4.4)(x - 0)
= (1/4.4)x + 1/4.4
Simplifying, we have:
y = (1/4.4)x + 1/4.4 + 1
= (1/4.4)x + 1/4.4 + 4/4.4
= (1/4.4)x + 5/4.4
= (1/4.4)x + 1.1364
Now, let's find the slope of the second line:
m2 = (y2 - y1) / (x2 - x1)
= (0 - 3) / (5.3 - 1.9)
= -3 / 3.4
So, the equation of the second line can be written as:
(y - 3) = (-3/3.4)(x - 1.9)
= (-3/3.4)x + 3/3.4 + 5.7/3.4
= (-3/3.4)x + 3/3.4 + 1.6765
= (-3/3.4)x + 1.6765
To find the intersection point, we can set the two equations equal to each other:
(1/4.4)x + 1.1364 = (-3/3.4)x + 1.6765
Simplifying, we have:
(1/4.4 + 3/3.4)x = 1.6765 - 1.1364
Combining the terms on the right side, we get:
(1/4.4 + 3/3.4)x = 0.5401
Now, we can solve for x:
(7.7/17.6)x = 0.5401
Dividing both sides by 7.7/17.6, we get:
x = (0.5401)/(7.7/17.6)
≈ 3.81
Now, we can substitute this value back into one of the equations to find y. Let's use the equation of the first line:
y = (1/4.4)(3.81) + 1.1364
≈ 1.98
So, the intersection point is approximately (3.81, 1.98).
To find the intersection point of two lines, we need to solve the system of equations formed by the equations of the two lines.
Step 1: Find the equation of the first line.
The equation of a line passing through two points (x₁, y₁) and (x₂, y₂) can be found using the slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
Using the points (0, 1) and (4.4, 2), we can find the slope (m₁):
m₁ = (y₂ - y₁) / (x₂ - x₁)
= (2 - 1) / (4.4 - 0)
= 1 / 4.4
Now, we can substitute one of the points (0, 1) and the slope (1 / 4.4) into the slope-intercept form to find the equation of the first line:
y = (1 / 4.4)x + b
Step 2: Find the value of b.
Substituting the coordinates (0, 1) into the equation, we can solve for b:
1 = (1 / 4.4)(0) + b
1 = b
So the equation of the first line is:
y = (1 / 4.4)x + 1
Step 3: Find the equation of the second line.
Using the points (1.9, 3) and (5.3, 0), we can find the slope (m₂) in the same way as before:
m₂ = (y₂ - y₁) / (x₂ - x₁)
= (0 - 3) / (5.3 - 1.9)
= -3 / 3.4
Substituting one of the points (1.9, 3) and the slope (-3 / 3.4) into the slope-intercept form, we can find the equation of the second line:
y = (-3 / 3.4)x + b
Step 4: Find the value of b.
Substituting the coordinates (1.9, 3) into the equation, we can solve for b:
3 = (-3 / 3.4)(1.9) + b
3 = -5.5882 + b
b = 8.5882
So the equation of the second line is:
y = (-3 / 3.4)x + 8.5882
Step 5: Solve the system of equations.
Now that we have the equations of both lines, we can solve the system of equations to find the intersection point. We can do this by setting the two y-values equal to each other and solving for x.
(1 / 4.4)x + 1 = (-3 / 3.4)x + 8.5882
Rearranging the equation:
(1 / 4.4)x + (3 / 3.4)x = 8.5882 - 1
[(1 / 4.4) + (3 / 3.4)]x = 7.5882
[(3 / 13/10) + (4 / 13/10)]x = 7.5882
[(7 / 13/10)]x = 7.5882
(10 / 13)(7x) = 7.5882
Multiplying both sides by (13/10):
7x = 7.5882 * (13/10)
Dividing both sides by 7:
x = (7.5882 * (13/10)) / 7
Calculating the value of x:
x = 9.8745
Step 6: Find the value of y.
Using the value of x, we substitute it into either of the lines to find the value of y. Let's use the first line:
y = (1 / 4.4)(9.8745) + 1
Calculating the value of y:
y = 2.2438
Therefore, the intersection point of the two lines is approximately (9.9, 2.2) rounded to the nearest tenth.