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?

Accepted Answer

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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