Determine the Equation of a Linear Function - Pre-Calculus

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Question

What is the equation of the line that passes through the points and ?

Express your answer in form.

Answer

First, we need to compute , the slope. We can do this with the slope formula

$m = \frac{y_2-y_1}{x_2-x_1}$ , sometimes called "rise over run"

$m = \frac{7-2}{4-0} = \frac{5}{4}$

So we now have

$y = \frac{5}{4}x +b$

Now in order to solve for we substitute one of our points into the equation we found. It doesn't matter which point we use, so we'll use .

We then have:

$2 = \frac{5}{4}0 +b$

Which becomes .

Hence we take our found value for and plug it back into $y= \frac{5}{4}x+b$ to get

$y =\frac{5}{4}x+2$ .

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Question

Find the equation of the line with slope that passes through the point $\left(\frac{5}{4},3\right)$ .

Express your answer in form.

Answer

Since our slope is , we can plug it into right away giving .

To solve for , we plug our given point $\left(\frac{5}{4},3\right)$ in for and giving $3 = 2\frac{5}{4} +b$ .

This will simplify to $3-\frac{10}{4} = b$ , or $\frac{1}{2} = b$ after subtracting the fraction.

Hence our answer is $y = 2x + \frac{1}{2}$ after plugging our values for and in.

Compare your answer with the correct one above

Question

Determine the equation of a line that passes through the points and .

Answer

Linear functions follow the form , where m is the slope and b is the y intercept. We can determine the equation of a linear function when we have the slope and a y intercept which is a starting point for drawing our line. If we need the equation of a line that passes through two points we use the point slope equation:

where the variables with numbers next to them correspond to places where you input the x and y from a single (same) coordinate pair.

Since we have two points, we can compute the slope of a line that passes between them. .

.

Now use the point slope formula:

distribute the right side

add 6 to both sides

which is the final answer.

Take a look at the coordinate pairs. Was it necessary to to use the point slope formula? The answer is no, since the coordinate pair (0,5) is already a y intercept. So all that was necessary was to compute the slope.

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Question

Suppose Bob has 4 candies. He then earns candies at a rate of 14 candies per week. Which of the following formulas is the most reasonable rate to know how many candies Bob has on any given day?

Answer

Bob starts out with 4 candies. Write the equation.

Every week, he earns 14 candies. Every week has a total of 7 days. Divide the amount of candies by the number of days to determine the number of candies Bob earn in a day.

$\frac{14: \textup{candies}}{7: \textup{day}}=\frac{2: \textup{candies}}{1: \textup{day}}$

Bob then earns 2 candies per day. Let represent the number of candies per day. Finish the incomplete equation.

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Question

After one hour of working, Billy has three dollars. At the end of eight hours, Billy has fifty dollars. Choose the most correct linear function representing this scenario if represents time, and represents dollars.

Answer

To solve for the equation of this problem, we will need 2 points. Let the point be defined as , where is time in hours and represents the dollars Billy has earned.

Billy starts with three dollars in the first hour. Write the point.

At the end of eight hours, Billy has fifty dollars. Write the second point.

Write the slope-intercept formula.

Write slope formula.

$m=\frac{y_2-y_1}{x_2-x_1}= \frac{50-3}{8-1}=\frac{47}{7}$

Plug a point and the given slope to the slope-intercept formula to solve for the y-intercept.

$3=(\frac{47}{7})(1)+b$

$b=\frac{21}{7}-\frac{47}{7}= -\frac{26}{7}$

Substitute this and the slope back into the slope-intercept equation.

$y=\frac{47}{7}x-\frac{26}{7}$

Since Billy's earning is dependent on the time he works, rewrite the equation so that time is the independent variable, and amount earned is the dependent.

$d=\frac{47}{7}t-\frac{26}{7}$

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Question

What equation is perpendicular to and passes throgh ?

Answer

First find the reciprocal of the slope of the given function.
$m_2 = -\frac{1}{m_1}$

$m_2 = -\frac{1}{5}$

The perpendicular function is:

$y = -\frac{1}{5}x+c$

Now we must find the constant, , by using the given point that the perpendicular crosses.

$10 = -\frac{1}{5}(6)+c$

solve for :

$c = \frac{56}{5}$

$y = -\frac{1}{5}x+\frac{56}{5}$

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Question

What equation is perpendicular to $y = -\frac{2}{5}x+3$ and passes throgh ?

Answer

First find the reciprocal of the slope of the given function.
$m_2 = -\frac{1}{m_1}$

$m_2 = \frac{5}{2}$

The perpendicular function is:

$y = \frac{5}{2}x+c$

Now we must find the constant, , by using the given point that the perpendicular crosses.

$4 = \frac{5}{2}(2)+c$

solve for :

$y = \frac{5}{2}x - 1$

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Question

What equation is perpendicular to $y = \frac{3}{7}x+12$ and passes throgh ?

Answer

First find the reciprocal of the slope of the given function.
$m_2 = -\frac{1}{m_1}$

$m_2 = -\frac{7}{3}$

The perpendicular function is:

$y = -\frac{7}{3}x+c$

Now we must find the constant, , by using the given point that the perpendicular crosses.

$4 = -\frac{7}{3}(5)+c$

solve for :

$c = \frac{47}{3}$

$y = -\frac{7}{3}x + \frac{47}{3}$

Compare your answer with the correct one above

Question

What is the equation of the line that passes through the points and ?

Express your answer in form.

Answer

First, we need to compute , the slope. We can do this with the slope formula

$m = \frac{y_2-y_1}{x_2-x_1}$ , sometimes called "rise over run"

$m = \frac{7-2}{4-0} = \frac{5}{4}$

So we now have

$y = \frac{5}{4}x +b$

Now in order to solve for we substitute one of our points into the equation we found. It doesn't matter which point we use, so we'll use .

We then have:

$2 = \frac{5}{4}0 +b$

Which becomes .

Hence we take our found value for and plug it back into $y= \frac{5}{4}x+b$ to get

$y =\frac{5}{4}x+2$ .

Compare your answer with the correct one above

Question

Find the equation of the line with slope that passes through the point $\left(\frac{5}{4},3\right)$ .

Express your answer in form.

Answer

Since our slope is , we can plug it into right away giving .

To solve for , we plug our given point $\left(\frac{5}{4},3\right)$ in for and giving $3 = 2\frac{5}{4} +b$ .

This will simplify to $3-\frac{10}{4} = b$ , or $\frac{1}{2} = b$ after subtracting the fraction.

Hence our answer is $y = 2x + \frac{1}{2}$ after plugging our values for and in.

Compare your answer with the correct one above

Question

Determine the equation of a line that passes through the points and .

Answer

Linear functions follow the form , where m is the slope and b is the y intercept. We can determine the equation of a linear function when we have the slope and a y intercept which is a starting point for drawing our line. If we need the equation of a line that passes through two points we use the point slope equation:

where the variables with numbers next to them correspond to places where you input the x and y from a single (same) coordinate pair.

Since we have two points, we can compute the slope of a line that passes between them. .

.

Now use the point slope formula:

distribute the right side

add 6 to both sides

which is the final answer.

Take a look at the coordinate pairs. Was it necessary to to use the point slope formula? The answer is no, since the coordinate pair (0,5) is already a y intercept. So all that was necessary was to compute the slope.

Compare your answer with the correct one above

Question

Suppose Bob has 4 candies. He then earns candies at a rate of 14 candies per week. Which of the following formulas is the most reasonable rate to know how many candies Bob has on any given day?

Answer

Bob starts out with 4 candies. Write the equation.

Every week, he earns 14 candies. Every week has a total of 7 days. Divide the amount of candies by the number of days to determine the number of candies Bob earn in a day.

$\frac{14: \textup{candies}}{7: \textup{day}}=\frac{2: \textup{candies}}{1: \textup{day}}$

Bob then earns 2 candies per day. Let represent the number of candies per day. Finish the incomplete equation.

Compare your answer with the correct one above

Question

After one hour of working, Billy has three dollars. At the end of eight hours, Billy has fifty dollars. Choose the most correct linear function representing this scenario if represents time, and represents dollars.

Answer

To solve for the equation of this problem, we will need 2 points. Let the point be defined as , where is time in hours and represents the dollars Billy has earned.

Billy starts with three dollars in the first hour. Write the point.

At the end of eight hours, Billy has fifty dollars. Write the second point.

Write the slope-intercept formula.

Write slope formula.

$m=\frac{y_2-y_1}{x_2-x_1}= \frac{50-3}{8-1}=\frac{47}{7}$

Plug a point and the given slope to the slope-intercept formula to solve for the y-intercept.

$3=(\frac{47}{7})(1)+b$

$b=\frac{21}{7}-\frac{47}{7}= -\frac{26}{7}$

Substitute this and the slope back into the slope-intercept equation.

$y=\frac{47}{7}x-\frac{26}{7}$

Since Billy's earning is dependent on the time he works, rewrite the equation so that time is the independent variable, and amount earned is the dependent.

$d=\frac{47}{7}t-\frac{26}{7}$

Compare your answer with the correct one above

Question

What equation is perpendicular to and passes throgh ?

Answer

First find the reciprocal of the slope of the given function.
$m_2 = -\frac{1}{m_1}$

$m_2 = -\frac{1}{5}$

The perpendicular function is:

$y = -\frac{1}{5}x+c$

Now we must find the constant, , by using the given point that the perpendicular crosses.

$10 = -\frac{1}{5}(6)+c$

solve for :

$c = \frac{56}{5}$

$y = -\frac{1}{5}x+\frac{56}{5}$

Compare your answer with the correct one above

Question

What equation is perpendicular to $y = -\frac{2}{5}x+3$ and passes throgh ?

Answer

First find the reciprocal of the slope of the given function.
$m_2 = -\frac{1}{m_1}$

$m_2 = \frac{5}{2}$

The perpendicular function is:

$y = \frac{5}{2}x+c$

Now we must find the constant, , by using the given point that the perpendicular crosses.

$4 = \frac{5}{2}(2)+c$

solve for :

$y = \frac{5}{2}x - 1$

Compare your answer with the correct one above

Question

What equation is perpendicular to $y = \frac{3}{7}x+12$ and passes throgh ?

Answer

First find the reciprocal of the slope of the given function.
$m_2 = -\frac{1}{m_1}$

$m_2 = -\frac{7}{3}$

The perpendicular function is:

$y = -\frac{7}{3}x+c$

Now we must find the constant, , by using the given point that the perpendicular crosses.

$4 = -\frac{7}{3}(5)+c$

solve for :

$c = \frac{47}{3}$

$y = -\frac{7}{3}x + \frac{47}{3}$

Compare your answer with the correct one above

Question

What is the equation of the line that passes through the points and ?

Express your answer in form.

Answer

First, we need to compute , the slope. We can do this with the slope formula

$m = \frac{y_2-y_1}{x_2-x_1}$ , sometimes called "rise over run"

$m = \frac{7-2}{4-0} = \frac{5}{4}$

So we now have

$y = \frac{5}{4}x +b$

Now in order to solve for we substitute one of our points into the equation we found. It doesn't matter which point we use, so we'll use .

We then have:

$2 = \frac{5}{4}0 +b$

Which becomes .

Hence we take our found value for and plug it back into $y= \frac{5}{4}x+b$ to get

$y =\frac{5}{4}x+2$ .

Compare your answer with the correct one above

Question

Find the equation of the line with slope that passes through the point $\left(\frac{5}{4},3\right)$ .

Express your answer in form.

Answer

Since our slope is , we can plug it into right away giving .

To solve for , we plug our given point $\left(\frac{5}{4},3\right)$ in for and giving $3 = 2\frac{5}{4} +b$ .

This will simplify to $3-\frac{10}{4} = b$ , or $\frac{1}{2} = b$ after subtracting the fraction.

Hence our answer is $y = 2x + \frac{1}{2}$ after plugging our values for and in.

Compare your answer with the correct one above

Question

Determine the equation of a line that passes through the points and .

Answer

Linear functions follow the form , where m is the slope and b is the y intercept. We can determine the equation of a linear function when we have the slope and a y intercept which is a starting point for drawing our line. If we need the equation of a line that passes through two points we use the point slope equation:

where the variables with numbers next to them correspond to places where you input the x and y from a single (same) coordinate pair.

Since we have two points, we can compute the slope of a line that passes between them. .

.

Now use the point slope formula:

distribute the right side

add 6 to both sides

which is the final answer.

Take a look at the coordinate pairs. Was it necessary to to use the point slope formula? The answer is no, since the coordinate pair (0,5) is already a y intercept. So all that was necessary was to compute the slope.

Compare your answer with the correct one above

Question

Suppose Bob has 4 candies. He then earns candies at a rate of 14 candies per week. Which of the following formulas is the most reasonable rate to know how many candies Bob has on any given day?

Answer

Bob starts out with 4 candies. Write the equation.

Every week, he earns 14 candies. Every week has a total of 7 days. Divide the amount of candies by the number of days to determine the number of candies Bob earn in a day.

$\frac{14: \textup{candies}}{7: \textup{day}}=\frac{2: \textup{candies}}{1: \textup{day}}$

Bob then earns 2 candies per day. Let represent the number of candies per day. Finish the incomplete equation.

Compare your answer with the correct one above

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