Writing Linear Equations, Applications, & Linear Regression **This is a 2-page documenti ** Point Slope & Two Points: Write a linear equation in slope-intercept form with the given Information 1. slope = -6; passes through (-4,1) 2. slope = passes through (-5, -6) 3. passes through (-4, 11) and (2,8) 4. passes through (6, 3) and (14,-5) Linear Equation Applications 5. Troy’s truck has a 30 gallon gas tank and gets an average of 21 miles per gallon. a) Write an equation to represent the amount of b) If the tank is half full, approximately how many gas in Troy’s truck after driving a certain miles has Troy driven since last filling up his number of miles (assuming he starts with a full tank? tank). Define your variables. 6. Mikayla is training for a half-marathon. At the beginning of her training, she is able to run 2 miles without stopping. Her goal is to increase her distance by three quarters of a mile each week. a) Write an equation to represent the number of b) If a half-marathon (13.1 miles) is 12 weeks miles Mikayla is able to run after each week. after the start of Mikayla’s training program, Define your variables. will she be ready? ** This is a 2-page Point Slope & Two Points: Write a linear equation information. 1. slope = -6; passes through (-4,1)
The Correct Answer and Explanation is :
Linear Equations: Point-Slope & Two Points
1. Slope = -6; passes through (-4, 1):
We are given the slope ( m = -6 ) and the point (-4, 1). To write the equation of the line, we will use the point-slope form of a linear equation, which is:
[
y – y_1 = m(x – x_1)
]
Substituting the given values for ( m = -6 ), ( x_1 = -4 ), and ( y_1 = 1 ):
[
y – 1 = -6(x + 4)
]
Simplifying:
[
y – 1 = -6x – 24
]
Now, solve for ( y ):
[
y = -6x – 23
]
So, the linear equation is:
[
y = -6x – 23
]
2. Slope = (not provided); passes through (-5, -6):
It seems there was an incomplete statement in the question. The slope needs to be provided in order to write the equation. If you have the slope value, you can use the same method as the first problem.
3. Passes through (-4, 11) and (2, 8):
To write the equation of the line passing through two points, we first need to calculate the slope. The formula for the slope ( m ) is:
[
m = \frac{y_2 – y_1}{x_2 – x_1}
]
Substituting the points (-4, 11) and (2, 8):
[
m = \frac{8 – 11}{2 – (-4)} = \frac{-3}{6} = -\frac{1}{2}
]
Now, use the point-slope form with one of the points (let’s use (-4, 11)):
[
y – 11 = -\frac{1}{2}(x + 4)
]
Simplifying:
[
y – 11 = -\frac{1}{2}x – 2
]
Now solve for ( y ):
[
y = -\frac{1}{2}x + 9
]
So, the linear equation is:
[
y = -\frac{1}{2}x + 9
]
4. Passes through (6, 3) and (14, -5):
First, find the slope using the formula:
[
m = \frac{-5 – 3}{14 – 6} = \frac{-8}{8} = -1
]
Using the point-slope form with the point (6, 3):
[
y – 3 = -1(x – 6)
]
Simplifying:
[
y – 3 = -x + 6
]
Now solve for ( y ):
[
y = -x + 9
]
So, the linear equation is:
[
y = -x + 9
]
Linear Equation Applications
5. Troy’s Truck Gas Tank:
Troy’s truck has a 30-gallon gas tank and gets 21 miles per gallon. Let ( x ) represent the number of miles driven and ( y ) represent the amount of gas left in the tank.
The equation for the amount of gas left after driving ( x ) miles is:
[
y = 30 – \frac{x}{21}
]
b) If the tank is half full, approximately how many miles has Troy driven since the last fill-up?
If the tank is half full, ( y = 15 ) gallons. Substitute this value into the equation:
[
15 = 30 – \frac{x}{21}
]
Solving for ( x ):
[
\frac{x}{21} = 15 \quad \Rightarrow \quad x = 315
]
Troy has driven approximately 315 miles since the last fill-up.
6. Mikayla’s Marathon Training:
Mikayla can run 2 miles without stopping, and she plans to increase her distance by ( \frac{3}{4} ) mile each week. Let ( x ) represent the number of weeks and ( y ) represent the number of miles Mikayla can run after ( x ) weeks.
The equation for her training is:
[
y = 2 + \frac{3}{4}x
]
b) Will Mikayla be ready for the half-marathon (13.1 miles) after 12 weeks?
Substitute ( x = 12 ) into the equation:
[
y = 2 + \frac{3}{4}(12) = 2 + 9 = 11
]
After 12 weeks, Mikayla will be able to run 11 miles, which is less than the 13.1 miles required for a half-marathon. Therefore, she will not be ready by then.
Explanation:
The problems involve writing linear equations using either the point-slope form or the slope-intercept form. The key steps include calculating the slope between two points and then using the point-slope form ( y – y_1 = m(x – x_1) ) to find the equation of the line. For applications, the equation of a real-world situation is derived from the given information, such as miles per gallon or training progression, and then used to find specific values (e.g., miles driven or training progress).