Practice questions

First find the slope from the equation of the line. For that write the equation in slope intercept form:

$y=-\dfrac{5}{2}x-\dfrac{27}{2}$

Coefficient of $x$ is the slope.

Slope, $m_1=-\dfrac{5}{2}$

Slope of the perpendicular line is negative of the reciprocal of $m_1$.

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

Equation of the perpendicular line in slope-intercept form is

$y=m_2x+b$

$y=\dfrac{2}{5}x+b$

Now, you need to find $b$.

Use the point, $(-5,-2)$ to find $b$. Substituting $x=-5$ and $y=-2$ in the above equation, you get

$-2=\dfrac{2}{5}(-5)+b$

$-2=-2+b$

$b=0$

Substitute this value of $b$ back into the equation, you get the answer:

$y=\dfrac{2}{5}x$

You are given two points, $(x_1,y_1)=(-3, 7)$ and $(x_2,y_2)=(2, -1)$.

First find the slope:

$m=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{-1-7}{2-(-3)}=-\dfrac{8}{5}$

Use the following formula to find the equation:

$(y-y_1)=m(x-x_1)$

$y-7=-\dfrac{8}{5}(x-(-3))$

$y-7=-\dfrac{8}{5}(x+3)$

Now, you need to write this in standard form:

Multiply $5$ on both sides:

$5(y-7)=-8(x+3)$

Remove the parentheses by multiplication

$5y-35=-8x-24$

Keep the $x$ and the $y$ terms on the left and the numbers on the right.

$5y+8x=-24+35$

$8x+5y=11$

Use the elimination method to solve:

Multiply the first equation by $4$ and the second equation by $-9$

$36x+20y=-76$

$-36x+27y=-18$

Add the equations that will eliminate $x$ terms:

$47y=-94$

Dividing $47$:

$y=-2$

Substitute $y=-2$ in one of the original equations and find $y$. Substituting $y=-2$ in the first equation:

$9x+5\cdot (-2)=-19$

$9x-10=-19$

Solving for $x$:

$x=-1$

So, the solutions are $x=-1$ and $y=-2$. Solution as ordered pair, $(-1, -2)$.

Use the elimination method to solve:

Multiply the first equation by $3$:

$6x-21y=-60$

Add this equation with the second equation, you will get,

$0=-45$

This is not true. So, the equation is inconsistent and has no solution.

The linear function is

$C(t)=16+40t$

To find the amount of time for the total cost to reach $\$560$, take $C(t)=560$ and solve for $t$.

$16+40t =560$

Solving for $t$,

$t=13.6$ months

Let $x$ is the speed of Rosanna and $y$ the speed of Simone.

Rosanna is 2 mph slower:

$x=y-2$  Equation (1)

Use, $time =\dfrac{distance}{speed}$

Both takes same amount of time:

$\dfrac{5}{x}=\dfrac{8}{y}$

Cross multiplying:

$5y=8x$

Substitute, $y-2$ for $x$ from Equation (1):

$5y=8(y-2)$

$5y=8y-16$

Solving for y:

$y=\dfrac{16}{3}=5.33$

Substitute, this $y$ in Equation (1) to find $x$:

$x=5.33-2=3.33$

Thus, the speed of Rosanna is 3.33 mph and that of Simone is 5.33 mph.

Use the following formula:

$\dfrac{1}{t}=\dfrac{1}{t_1}+\dfrac{1}{t_2}$

where $t_1$ and $t_2$ respectively are the time taken by the well and the spring to fill the pool if they work alone.

And, $t$ is the time when both work together.

It is given, $t=3$ hours

You need to find $t_2$. So, take $t_2=x$

The well takes 8 hours less than the spring:

$t_1=t_2-8=x-8$

Substituting the values of $t$, $t_1$ and $t_2$ in the equation:

$\dfrac{1}{3}=\dfrac{1}{x-8}+\dfrac{1}{x}$   Equation (1).

LCD of this equation is  $3(x-8)x.$

Multiply the LCD in each term, you will get

$(x-8)x=3x+3(x-8)$

$x^2-8x=3x+3x-24$

$x^2-14x+24=0$

Solve this quadratic equation by factoring:

$(x-12)(x-2)=0$

By zero-product rule:

$x=12$ or $x=2$

There are two times, only one should be the correct one.

$x=2$ makes the time, $t_1$ negative. So, this cannot be a solution.

So, the solution is $x=12$.

Thus, the spring will take 12 hours to fill the pool alone.

Take the expression inside the radical sign and set that $\ge0$:

$4-9x\ge 0$

Solve for $x$:

Subtract $-4$

$-9x\ge -4$

Divide $-9$, dividing a negative number will change the inequality sign:

$x\le \dfrac{-4}{-9}$

$x\le \dfrac{4}{9}$

Now, you can write the domain in interval notation or can graph on a number line.

In interval notation, $\left(-\infty, \dfrac{4}{9}\right]$

The function is a rational function. Take the denominator and set that equal to zero:

$x^2+12x+35=0$

Solve this quadratic equation by factoring:

$(x+7)(x+5)=0$

Using the zero-product rule:

$x+7=0$ or $x+5=0$

$x=-7$ or $x=-5$

These are the excluded values. So, the domain is all the real numbers except $-7$ and $-5$.

We can write the domain in interval notation as

$(-\infty, -7)\cup (-7, -5) \cup (-5, \infty)$

Note that parentheses exclude the two numbers, and you should write the smaller number first.

Let $x$ and $y$ are the amount of 25% and 35% alcohol respectively mixed together.

Total mixture is $20$ gallon:

$x+y=20$   Equation (1)

Looking at the amount of salt in each solution mixed and in the total mixture of 20 L:

$0.25x+0.35y=0.32*20$

Multiplying $100$ to remove the decimal point:

$25x+35y=640$   Equation (2)

Solve this and the first equation by elimination method:

Multiply equation (1) with $-25$:

$-25x-25y=-500$

Add this and equation (2):

$10y=140$

$y=14$

Now, substitute this in eqn.(1) to find $x$:

$x+14=20$

Solving for $x$:

$x=6$

So, to get the desired mixture, you need to mix 6 gallons of 25% and 14 gallons of 35% alcohol solution.

Let $x$ and $y$ are the numbers of \$8.50 and $9.75 brushes respectively.

So,

$x+y=45$  Equation (1)

Total price:

$8.5x+9.75y=398.75$

Multiply, 100 to remove the decimal point:

$850x+975y=39875$  Equation (2).

Solve equations (1) and (2) by elimination method:

Multiply $-850$ in equation (1)

$-850x-850y=-38250$

Add this equation with equation (2) that will eliminate $x$:

$125y=1625$

Divide 125:

$y=13$

Substitute this $y$ in equation (1) and find $x$:

$x+13=45$

or, $x=32$

Thus, the number of \$8.50 brush sold $=x=32$. And the number of \$9.75 brush sold $=y=13$.

Let $b$ is the speed of the boat in still water, and $w$ is the speed of the current:

Use the formula:

$speed =\dfrac{distance}{time}$

For upstream:

$b-w=\dfrac{72}{6}$

$b-w=12$  Equation (1)

For downstream:

$b+w=\dfrac{72}{4}$

$b+w=18$  Equation (2)

Solve Equations (1) and (2) by elimination method:

Add the two equations, you will eliminate $w$:

$2b=12+18$

$2b=30$

$b=15$

This is the speed of the boat in still water. You need to find speed of the current, $w$.

Substitute, the value of $b$ in equation (2) and find $w$:

$15+w=18$

$w=3$

Thus, the speed of the current is 3 mph.