Creating and Graphing Two Variable Equations - Algebra 2

1. What is the slope of the line through $(1,2)$ and $(5,10)$?

   $m=2$. Use slope formula: $m=\frac{10-2}{5-1}=\frac{8}{4}=2$.

2. Identify the $y$-intercept of $y=-2x+6$.

   $y=6$. Set $x=0$: $y=-2(0)+6=6$.

3. What is the equation for density $D$ given mass $m$ and volume $V$?

   $D=\frac{m}{V}$. Density equals mass per unit volume.

4. What is the slope-intercept form of a line in two variables?

   $y=mx+b$. Where $m$ is slope and $b$ is the $y$-intercept.

5. What is the equation of the line through $(2,5)$ with slope $-3$?

   $y-5=-3(x-2)$. Point-slope form using given point and slope.

6. What is the general form of a linear equation in two variables?

   $Ax+By=C$. Standard form where $A$, $B$, $C$ are constants and $A$, $B$ not both zero.

7. Identify the slope of the line given by $3x-2y=8$.

   $m=\frac{3}{2}$. Rewrite $3x-2y=8$ as $y=\frac{3}{2}x-4$ to find slope.

8. What is the equation stating that $y$ is $\frac{1}{n}$ of $x$?

   $y=\frac{x}{n}$. One quantity is a fraction of another.

9. What is the equation stating that $y$ is $n$ times $x$?

   $y=nx$. One quantity is a multiple of another.

10. What is the equation representing that the perimeter is $24$ for a rectangle with sides $x$ and $y$?

    $2x+2y=24$. Perimeter formula $P=2x+2y$ set equal to $24$.

11. What is the equation for average speed $v$ given distance $d$ and time $t$?

    $v=\frac{d}{t}$. Speed equals distance divided by time elapsed.

12. What is the equation of the circle centered at $(2,-1)$ with radius $3$?

    $(x-2)^2+(y+1)^2=9$. Standard circle form with center $(2,-1)$ and radius $3$.

13. What is the equation of a horizontal line passing through $y=k$?

    $y=k$. All points have the same $y$-coordinate $k$.

14. What is the point-slope form of a line through $(x_1,y_1)$ with slope $m$?

    $y-y_1=m(x-x_1)$. Uses a known point and slope to define the line.

15. What is the equation for perimeter $P$ of a rectangle with length $L$ and width $W$?

    $P=2L+2W$. Sum of all four sides of a rectangle.

16. What is the standard form of a parabola that opens up or down with vertex $(h,k)$?

    $y=a(x-h)^2+k$. Vertex form where $a$ determines width and direction of opening.

17. Identify the correct scale choice if $x$ values run from $0$ to $200$ on a $10$-tick axis.

    $20$ units per tick. Divide total range by number of intervals: $\frac{200}{10}=20$.

18. What is the equation of the line through $(0,-2)$ and $(3,4)$?

    $y=2x-2$. Slope is $\frac{4-(-2)}{3-0}=2$, use point-slope form.

19. What is the equation for converting Fahrenheit $F$ to Celsius $C$?

    $C=\frac{5}{9}(F-32)$. Temperature conversion from Fahrenheit to Celsius scale.

20. What is the meaning of the $x$-intercept of a graph in terms of solutions?

    The $x$-value where $y=0$. Point where graph crosses horizontal axis.

21. What is the equation for remaining money $R$ after spending $s$ from an initial $50$?

    $R=50-s$. Remaining amount decreases linearly with spending.

22. What is the equation for distance $d$ at constant speed $r$ over time $t$?

    $d=rt$. Distance equals rate times time for uniform motion.

23. What is the equation of the line with slope $4$ and $y$-intercept $-3$?

    $y=4x-3$. Direct substitution into slope-intercept form $y=mx+b$.

24. What is the equation for the sum of two numbers $x$ and $y$ being $S$?

    $x+y=S$. Two quantities add up to a given total $S$.

25. What is the equation for joint variation where $y$ varies jointly with $x$ and $z$?

    $y=kxz$. One variable proportional to product of two others.

26. Identify the $y$-intercept of the line given by $3x-2y=8$.

    $b=-4$. From slope-intercept form $y=\frac{3}{2}x-4$, $y$-intercept is $-4$.

27. What is the equation stating that $y$ is $n$ more than $x$?

    $y=x+n$. One quantity exceeds another by amount $n$.

28. What is the equation for total pay with hourly wage $15$ for $h$ hours plus bonus $20$?

    $P=15h+20$. Linear pay function with hourly rate plus fixed bonus.

29. What is the equation of a vertical line passing through $x=h$?

    $x=h$. All points have the same $x$-coordinate $h$.

30. What is the equation of the horizontal line through $(7,-4)$?

    $y=-4$. Horizontal lines have constant $y$-value.

31. What is the equation for combined variation where $y$ varies directly with $x$ and inversely with $z$?

    $y=\frac{kx}{z}$. One variable proportional to quotient of two others.

32. What is the meaning of the $y$-intercept of a graph in terms of solutions?

    The $y$-value where $x=0$. Point where graph crosses vertical axis.

33. What is the equation of the vertical line through $(-3,8)$?

    $x=-3$. Vertical lines have constant $x$-value.

34. What is the equation for inverse variation between $y$ and $x$ with constant $k$?

    $y=\frac{k}{x}$. Product $xy$ remains constant as one increases, other decreases.

35. What is the equation representing that $y$ varies directly with $x$ and $y=12$ when $x=3$?

    $y=4x$. Find $k$ using $12=4(3)$, so $k=4$.

36. What is the equation for profit $P$ in terms of revenue $R$ and cost $C$?

    $P=R-C$. Profit is revenue minus total costs.

37. What is the standard form of a parabola that opens left or right with vertex $(h,k)$?

    $x=a(y-k)^2+h$. Horizontal parabola with vertex $(h,k)$ and parameter $a$.

38. What is the equation for a taxi fare with base fee $3$ and $2$ dollars per mile $m$?

    $C=2m+3$. Linear cost function with fixed base fee and variable rate.

39. Identify the equation representing that the sum of angles $x$ and $y$ is $180$.

    $x+y=180$. Two angles that sum to $180$ degrees are supplementary.

40. What is the slope formula using two points $(x_1,y_1)$ and $(x_2,y_2)$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run between any two distinct points on a line.

41. What is the equation representing inverse variation if $y=5$ when $x=2$?

    $y=\frac{10}{x}$. Find $k$ using $5=\frac{k}{2}$, so $k=10$.

42. Identify the $x$-intercept of $y=-2x+6$.

    $x=3$. Set $y=0$ and solve: $0=-2x+6$, so $x=3$.

43. What does it mean for an ordered pair $(x,y)$ to satisfy an equation in two variables?

    It makes the equation true when substituted. The coordinates produce a true statement when plugged in.

44. What is the equation for direct variation between $y$ and $x$ with constant $k$?

    $y=kx$. Linear relationship through origin with constant ratio $k$.

45. What is the equation representing that the area is $30$ for a rectangle with sides $x$ and $y$?

    $xy=30$. Area formula $A=xy$ set equal to $30$.

46. What is the equation for revenue $R$ when price per unit is $p$ and units sold is $x$?

    $R=px$. Revenue equals price per unit times number of units sold.

47. What is the equation of the parabola with vertex $(1,-4)$ and $a=2$ opening upward?

    $y=2(x-1)^2-4$. Vertex form with $(h,k)=(1,-4)$ and $a=2$.

48. What is the equation for simple interest $I$ with principal $P$, rate $r$, and time $t$?

    $I=Prt$. Interest equals principal times rate times time period.

49. What is the standard form of a circle with center $(h,k)$ and radius $r$?

    $(x-h)^2+(y-k)^2=r^2$. Distance from center $(h,k)$ to any point on circle equals $r$.

50. What is the equation stating that $y$ is $n$ less than $x$?

    $y=x-n$. One quantity is smaller than another by amount $n$.

51. What is the equation of a line with slope $m$ passing through the origin?

    $y=mx$. Line through origin with slope $m$ and $y$-intercept $0$.

52. What is the equation for area $A$ of a rectangle in terms of length $L$ and width $W$?

    $A=LW$. Product of adjacent sides gives rectangular area.

53. What is the equation for total cost $C$ with fixed fee $f$ and per-unit cost $p$ for $x$ units?

    $C=px+f$. Total cost includes fixed fee plus variable cost per unit.

54. What is the equation for converting Celsius $C$ to Fahrenheit $F$?

    $F=\frac{9}{5}C+32$. Temperature conversion from Celsius to Fahrenheit scale.

55. What is the equation of the parabola with vertex $ (1,-4) $ and $a=2$ opening upward?

    $y=2(x-1)^2-4$. Vertex form with $ (h,k)=(1,-4) $ and $a=2$.

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