Equations With Two Variables - SAT Math

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

   $Ax + By = C$. Standard form where A, B, and C are constants.

2. Find the y-intercept of the equation $2x + 3y = 6$.

   $(0, 2)$. Set $x = 0$: $3y = 6$, so $y = 2$.

3. What is the slope of the line $y = -5x + 2$?

   $-5$. Coefficient of $x$ in slope-intercept form.

4. What is the formula for converting point-slope form to slope-intercept form?

   $y - y_1 = m(x - x_1)$ to $y = mx + b$. Expand point-slope and solve for $y$.

5. Find the equation of the line with undefined slope through $(4, -1)$.

   $x = 4$. Undefined slope means vertical line.

6. What is the x-coordinate of the point where the line $y = 2x + 3$ crosses the x-axis?

   $-\frac{3}{2}$. Set $y = 0$: $0 = 2x + 3$, so $x = -\frac{3}{2}$.

7. What is the standard form of a linear equation in two variables?

   $Ax + By = C$. General form with integer coefficients $A$, $B$, and $C$.

8. Find the x-intercept of the line $3x + 4y = 12$.

   The x-intercept is $x = 4$.. Set $y = 0$ and solve for $x$.

9. What is the slope of the line that is horizontal?

   0. Horizontal lines have zero rise over run.

10. What is the equation of a vertical line passing through $(5, -2)$?

    $x = 5$. Vertical lines have constant $x$-values.

11. Convert the equation $2x + 3y = 6$ to slope-intercept form.

    $y = -\frac{2}{3}x + 2$. Isolate $y$ by subtracting $2x$ and dividing by $3$.

12. What is the formula for the slope of a line given points $ ($x_1$, $y_1$) $ and $ ($x_2$, $y_2$) $?

    Slope $m = \frac{y_2 - y_1}{x_2 - x_1}$. Rise over run between two points.

13. What is the equation of a horizontal line passing through $(3, 7)$?

    $y = 7$. Horizontal lines have constant $y$-values.

14. Solve for $y$: $4x - 2y = 8$.

    $y = 2x - 4$. Isolate $y$ by subtracting $4x$ and dividing by $-2$.

15. What is the standard form of a linear equation with two variables?

    $Ax + By = C$. General linear form where $A$, $B$, and $C$ are constants.

16. Convert $y = -2x + 5$ to standard form.

    $2x + y = 5$. Move $x$-term to left: $2x + y = 5$.

17. What is the slope of the line $y = -3$?

    0. Horizontal lines have zero slope.

18. Find the equation of a line with slope $0$ through $(-2, 5)$.

    $y = 5$. Zero slope creates horizontal line.

19. What is the y-coordinate of the point where the line $2x + 5y = 10$ crosses the y-axis?

    $2$. Set $x = 0$: $5y = 10$, so $y = 2$.

20. Find the x-intercept of the equation $4x - y = 8$.

    $(2, 0)$. Set $y = 0$: $4x = 8$, so $x = 2$.

21. Convert $y = 3x - 2$ to point-slope form through $(1, 1)$.

    $y - 1 = 3(x - 1)$. Use point $(1,1)$ in point-slope formula.

22. Find the x-intercept of the equation $3x + 2y = 12$. What is it?

    $x = 4$. Set $y = 0$ and solve for $x$.

23. Find the y-intercept of the line $3x + 6y = 12$.

    $(0, 2)$. Set $x = 0$: $6y = 12$, so $y = 2$.

24. What is the slope of the line $x = 7$?

    Undefined. Vertical lines have undefined slope.

25. Find the slope of the line $y = -\frac{3}{2}x + 4$.

    $-\frac{3}{2}$. Coefficient of $x$ in slope-intercept form.

26. Find the equation of a line with slope $3$ and y-intercept $-4$.

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

27. What is the slope of a line perpendicular to $y = 4x + 7$?

    $-\frac{1}{4}$. Perpendicular slopes are negative reciprocals.

28. What is the formula for a line in point-slope form?

    $y - y_1 = m(x - x_1)$. Uses known point $(x_1, y_1)$ and slope $m$.

29. Identify the slope in the equation $y = mx + b$.

    $m$. Coefficient of $x$ represents the rate of change.

30. Find the slope of the line given by $2x - 3y = 6$.

    $\frac{2}{3}$. Rearrange to $y = \frac{2}{3}x - 2$ to identify slope.

31. State the condition for two lines to be parallel.

    Slopes are equal. Parallel lines have identical slopes but different intercepts.

32. Find the slope of the line passing through $(2, 3)$ and $(4, 7)$.

    $2$. Use slope formula: $\frac{7-3}{4-2} = \frac{4}{2} = 2$.

33. State the formula for the slope between points $(x_1, y_1)$ and $(x_2, y_2)$.

    $m = \frac{y_2 - y_1}{x_2 - x_1}$. Rise over run: change in y divided by change in x.

34. Find the x-intercept of the line $y = 4x - 8$.

    $(2, 0)$. Set $y = 0$: $0 = 4x - 8$, so $x = 2$.

35. What is the slope of the line that is vertical?

    Undefined. Vertical lines have no finite slope.

36. State the formula for the slope of a line between points $(x_1, y_1)$ and $(x_2, y_2)$.

    $m = \frac{y_2 - y_1}{x_2 - x_1}$. Rise over run formula for any two points.

37. What is the value of $x$ in the system: $2x + 3y = 6$, $x - y = 1$?

    $x = 3$. Substitute and solve the system of equations.

38. Find the solution for $x$ if $y = 3$ in the equation $2x + y = 7$.

    $x = 2$. Substitute $y = 3$ and solve: $2x + 3 = 7$.

39. Convert the equation $3x + 4y = 12$ to slope-intercept form.

    $y = -\frac{3}{4}x + 3$. Solve for $y$: $4y = -3x + 12$, then $y = -\frac{3}{4}x + 3$.

40. Find the equation of the line with slope $-2$ passing through $(1, 4)$.

    $y = -2x + 6$. Use point-slope: $y - 4 = -2(x - 1)$, then solve.

41. 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.

42. What is the y-intercept in the equation $y = mx + b$?

    $b$. Constant term where line crosses y-axis.

43. Convert the point-slope equation $y - 2 = 3(x + 1)$ to standard form.

    $3x - y = -5$. Expand and rearrange to $Ax + By = C$ form.

44. Convert the equation $2x + 3y = 6$ to slope-intercept form.

    $y = -\frac{2}{3}x + 2$. Solve for $y$ by isolating it on one side.

45. Find the equation of a line perpendicular to $y = 2x + 1$ through $(3, 2)$.

    $y = -\frac{1}{2}x + \frac{7}{2}$. Perpendicular slope $-\frac{1}{2}$, use point-slope form.

46. What form is the equation $y - 3 = 2(x - 1)$ in?

    Point-slope form. Form $y - y_1 = m(x - x_1)$ with known point.

47. What is the slope of a line parallel to $y = \frac{1}{2}x - 3$?

    $\frac{1}{2}$. Parallel lines have identical slopes.

48. Does the equation $y = 2x + 3$ represent a function? Yes or No.

    Yes. Each x-value has exactly one y-value.

49. Identify the form of $2x + 3y = 6$.

    Standard form. Form $Ax + By = C$ with integer coefficients.

50. Identify the slope in the equation $y = 3x + 7$. What is it?

    3. In $y = mx + b$ form, the coefficient of $x$ is the slope.

51. Identify the y-intercept in the equation $y = mx + b$.

    The y-intercept is $b$. The constant term where the line crosses the y-axis.

52. What is the y-intercept of the equation $y = -4x + 9$?

    9. In $y = mx + b$ form, $b$ is the y-intercept.

53. Solve for $y$: $4x - 2y = 8$.

    $y = 2x - 4$. Divide both sides by -2 to isolate $y$.

54. State the formula for the slope of a line between points $ ($x_1$, $y_1$) $ and $ ($x_2$, $y_2$) $.

    $m = \frac{y_2 - y_1}{x_2 - x_1}$. Rise over run formula for any two points.

55. Find the equation of a line parallel to $y = -\frac{1}{3}x + 2$ through $(6, 4)$.

    $y = -\frac{1}{3}x + 6$. Same slope, different y-intercept through given point.

56. Which form is $y = mx + b$ called?

    Slope-intercept form. Shows slope $m$ and y-intercept $b$ directly.

57. Determine the slope of the line $y = -5x + 3$.

    The slope is $-5$. The coefficient of $x$ in slope-intercept form.

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

    $(0, 4)$. Set $x = 0$: $2y = 8$, so $y = 4$.

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