Q: Simplify the fraction: $\dfrac{10xy + 15y - 4x - 6}{10x^2 + 11x - 6}$

$\dfrac{5y-2}{5x-2}$, where $x \neq -\dfrac{3}{2}$ or $\dfrac{2}{5}$

Question 1

Solve the following system of equations for $x$ and $y$:

$\begin{cases}
3x + 3y = 14 \
5x + 6y = 24
\end{cases}$

Accepted Answer

$x = 4,\ y = \dfrac{2}{3}$

Question 2

Solve the following system of equations for $x$ and $y$:

$\begin{cases}
3x + 6y = 30 \
5x + 4y = 32
\end{cases}$

Accepted Answer

$x = 4,\ y = 3$

Question 3

Solve for $x$:

$\dfrac{1}{x-2} + \dfrac{1}{4} = \dfrac{4}{x^2-4}$

Accepted Answer

$x = -6$

Question 4

Simplify the complex fraction:

$\dfrac{3r+2}{r+1} \div \left( \dfrac{1}{r+1} + \dfrac{5}{r-1} \right)$

Accepted Answer

$\dfrac{r-1}{2}$

Question 5

Solve for $r$:

$\dfrac{5}{r-2} - \dfrac{1}{2} = \dfrac{3}{r+1}$

Accepted Answer

$r = -3$ and $r = 8$

Question 6

Simplify the expression:

$\dfrac{1}{m^2 + 5m + 6} + \dfrac{2}{m^2 + m - 6}$

Accepted Answer

$\dfrac{3m+2}{(m+3)(m+2)(m-2)}$

Question 7

Simplify the fraction:

$\dfrac{10xy + 15y - 4x - 6}{10x^2 + 11x - 6}$

Accepted Answer

$\dfrac{5y-2}{5x-2}$, where $x 
eq -\dfrac{3}{2}$ or $\dfrac{2}{5}$

Question 8

A company is analyzing the cost of producing widgets. The cost equation is $5x - 3y = -12$, where $x$ is the number of widgets and $y$ is the total cost in dollars. What is the slope of this equation?

Question 9

What is the slope of the line $y = 4$? What does the graph of this line look like?

Accepted Answer

Slope is $0$; the line is horizontal

Question 10

Find the equation of the line passing through the points $(2, -5)$ and $(1, 7)$. Write your answer in both slope-intercept and standard forms.

Accepted Answer

$y = -12x + 19$ and $12x + y = 19$

Question 11

Solve for $x$:

$3x(x+2) - 7 = 2x^2 + 7x + 5$

Accepted Answer

$x = -3$ and $x = 4$

Question 12

A biologist is mixing two solutions. One solution contains $x$ grams of chemical A and $y$ grams of chemical B. If $3x + 3y = 14$ and $5x + 6y = 24$, how many grams of chemical A are in the mixture?

Accepted Answer

$4$ grams

Question 13

A horizontal line passes through the point $(0, 4)$. Which equation represents this line?

Accepted Answer

$y = 4$

Question 14

A company’s revenue is modeled by $y = mx + b$. If the revenue does not change as the number of units sold increases, what is the value of $m$?

Accepted Answer

$m = 0$

Question 15

Which of the following is the correct standard form of the equation for a line with slope $-12$ and $y$-intercept $19$?

Accepted Answer

$12x + y = 19$

Question 16

A student simplifies $\dfrac{1}{m^2 + 5m + 6} + \dfrac{2}{m^2 + m - 6}$ and gets $\dfrac{3m+2}{(m+3)(m+2)(m-2)}$. For which values of $m$ is this expression undefined?

Accepted Answer

$m = -3,\ m = -2,\ m = 2$

Question 17

A vertical line passes through the point $(5, 0)$. What is the equation of this line?

Accepted Answer

$x = 5$

Question 18

If the equation $5x - 3y = -12$ is rewritten in slope-intercept form, what is the $y$-intercept?

Accepted Answer

$y$-intercept is $4$

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