Creating and Graphing Two Variable Equations - Algebra 1

1. What is the equation for a line with slope $-2$ and $y$-intercept $5$?

   $y=-2x+5$. Substitute slope and $y$-intercept into $y=mx+b$.

2. What is the equation for the area $A$ of a circle with radius $r$?

   $A=\pi r^2$. Space enclosed by the circle.

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

   $y=kx$. Direct variation passes through the origin.

4. What is the equation in slope-intercept form for $-3x+6y=12$?

   $y=\frac{1}{2}x+2$. Divide by $6$: $y=\frac{1}{2}x+2$.

5. What is the equation for cost $C$ if fixed cost is $50$ and variable cost is $6$ per unit $x$?

   $C=6x+50$. Fixed cost plus variable cost per unit.

6. What is the equation for taxi cost $C$ with base fare $4$ and $2.5$ dollars per mile $m$?

   $C=2.5m+4$. Base fare plus cost per mile.

7. What is the equation for the circumference $C$ of a circle with radius $r$?

   $C=2\pi r$. Distance around the circle's edge.

8. Identify the equation modeling "$y$ varies directly with $x$" and $y=12$ when $x=3$.

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

9. What is the meaning of $b$ in $y=mx+b$ in terms of an initial value?

   $b$ is the initial value when $x=0$. The $y$-intercept represents the starting value.

10. What is the equation for a constant difference relationship where $y$ is $7$ less than $x$?

    $y=x-7$. Subtract $7$ from any $x$-value to get $y$.

11. Which $x$-values are most efficient to graph $y=2x-1$ quickly by a table?

    Use small integers such as $x=-1,0,1$. Easy values for mental calculation.

12. 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.

13. What scale choice is appropriate when graphing points with coordinates up to $20$?

    Use a consistent scale such as $1$ or $2$ units per grid mark. Prevents crowding and ensures readability.

14. What is the equation of the line through $(2,-1)$ with slope $2$?

    $y=2x-5$. Use point-slope form with $(2,-1)$ and $m=2$.

15. What is the equation of a vertical line passing through $x=3$?

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

16. What is the equation for profit $P$ if revenue is $R$ and cost is $C$?

    $P=R-C$. Revenue minus total costs.

17. Identify the equation modeling "$y$ varies directly with $x$" and $y=-10$ when $x=5$.

    $y=-2x$. Find $k$: $-10=k(5)$, so $k=-2$.

18. Identify the independent variable in the model $C=3v+10$ where $v$ is number of visits.

    The independent variable is $v$. The input variable that can be chosen.

19. What is the $y$-intercept of a graph, stated as a condition on $x$?

    The $y$-intercept occurs where $x=0$. Set $x=0$ and solve for $y$.

20. What is the $y$-intercept of the line $3x+2y=8$?

    $y=4$. Set $x=0$ in $3x+2y=8$ to get $y=4$.

21. Identify the dependent variable in the model $C=3v+10$ where $v$ is number of visits.

    The dependent variable is $C$. The output variable that depends on $v$.

22. What is the $y$-intercept of a line written in the form $y=mx+b$?

    The $y$-intercept is $b$. The constant term gives the $y$-intercept.

23. What is the equation for a constant difference relationship where $y$ is $5$ more than $x$?

    $y=x+5$. Add $5$ to any $x$-value to get $y$.

24. What is the equation of a horizontal line passing through $y=-2$?

    $y=-2$. Horizontal lines have constant $y$-coordinate.

25. What is the equation in slope-intercept form for $2x+y=7$?

    $y=-2x+7$. Solve for $y$: $y=-2x+7$.

26. What is the $x$-intercept of a graph, stated as a condition on $y$?

    The $x$-intercept occurs where $y=0$. Set $y=0$ and solve for $x$.

27. What is the standard form of a linear equation in two variables used in Algebra $1$?

    $Ax+By=C$. General linear form with integer coefficients.

28. What is the equation for revenue $R$ when each item costs $8$ dollars and $x$ items are sold?

    $R=8x$. Price per item times number sold.

29. What is the equation for total pay $P$ with hourly wage $15$ and $h$ hours worked?

    $P=15h$. Hourly wage times hours worked.

30. What does the variable $y$ typically represent on a coordinate plane when graphing an equation?

    $y$ represents the dependent variable (vertical axis). It depends on the $x$-value chosen.

31. What is the meaning of the slope $m$ in $y=mx+b$ as a rate of change?

    $m$ is the change in $y$ per $1$ unit change in $x$. Slope measures the rate of change.

32. What is the slope of the line through $(0,2)$ and $(3,11)$?

    $m=3$. Use slope formula: $m=\frac{11-2}{3-0}=3$.

33. What is the equation for a line with slope $4$ and $y$-intercept $-3$?

    $y=4x-3$. Substitute slope and $y$-intercept into $y=mx+b$.

34. What is the equation for total distance $d$ if you start with $d_0$ and travel at speed $v$ for time $t$?

    $d=d_0+vt$. Initial distance plus distance traveled.

35. What is the first step to graph an equation in two variables by making a table?

    Choose $x$-values and compute the corresponding $y$-values. Create ordered pairs to plot.

36. Identify the equation for Celsius $C$ in terms of Fahrenheit $F$ using a linear conversion.

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

37. What is the equation for total cost $C$ with $10$ dollar membership fee plus $3$ dollars per visit $v$?

    $C=3v+10$. Fixed fee plus cost per visit.

38. What is the slope-intercept form of a linear equation in two variables?

    $y=mx+b$. Standard form with slope $m$ and $y$-intercept $b$.

39. Find the missing value: If $(2,y)$ is on $y=3x-4$, what is $y$?

    $y=2$. Substitute $x=2$: $y=3(2)-4=2$.

40. What does it mean for an ordered pair $(x,y)$ to be a solution of an equation in two variables?

    $(x,y)$ makes the equation true when substituted. The coordinates satisfy the equation exactly.

41. What is the equation for the area $A$ of a rectangle with length $L$ and width $W$?

    $A=LW$. Product of length and width.

42. What is the slope of the line through $(2,5)$ and $(6,1)$?

    $m=-1$. Use slope formula: $m=\frac{1-5}{6-2}=-1$.

43. Find the missing value: If $(x,5)$ is on $y=2x+1$, what is $x$?

    $x=2$. Substitute $y=5$: $5=2x+1$, so $x=2$.

44. What labels should be included on a coordinate graph of $y$ versus $x$?

    Label axes with variable names and units (if given). Proper labeling makes graphs interpretable.

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

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

46. What is the equation of the line through $(1,4)$ with slope $-3$?

    $y=-3x+7$. Use point-slope form with $(1,4)$ and $m=-3$.

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

    $m=\frac{y_2-y_1}{x_2-x_1}$. Change in $y$ divided by change in $x$.

48. What is the equation for a proportional relationship between $y$ and $x$?

    $y=kx$ for some constant $k$. The ratio $\frac{y}{x}$ remains constant.

49. Identify the equation for Fahrenheit $F$ in terms of Celsius $C$ using a linear conversion.

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

50. What does the variable $x$ typically represent on a coordinate plane when graphing an equation?

    $x$ represents the independent variable (horizontal axis). It's the input value that can be chosen freely.

51. What is the equation for total amount $A$ after simple interest, using $A$, $P$, $r$, and $t$?

    $A=P+Prt$. Principal plus interest earned.

52. What is the $x$-intercept of the line $3x+2y=8$?

    $x=\frac{8}{3}$. Set $y=0$ in $3x+2y=8$ to get $x=\frac{8}{3}$.

53. What is the equation for total cost $C$ with a fixed fee $f$ and rate $r$ per unit $x$?

    $C=rx+f$. Fixed fee plus variable cost per unit.

54. What is the equation for distance $d$ traveled at constant speed $v$ for time $t$?

    $d=vt$. Distance equals rate times time.

55. Identify the slope and $y$-intercept in $y=-\frac{3}{4}x+6$.

    $m=-\frac{3}{4}$ and $b=6$. Coefficient of $x$ is slope, constant is $y$-intercept.

56. Identify the slope of the line given by $5x-10y=20$.

    $m=\frac{1}{2}$. Rewrite as $y=\frac{1}{2}x-2$; slope is $\frac{1}{2}$.

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