Table of contents

0. Review of Algebra(0)
1. Equations & Inequalities(0)
2. Graphs of Equations(0)
3. Functions(0)
4. Polynomial Functions(0)
5. Rational Functions(0)
6. Exponential & Logarithmic Functions(0)
7. Systems of Equations & Matrices(0)
8. Conic Sections(0)
9. Sequences, Series, & Induction(0)
10. Combinatorics & Probability(0)

7. Systems of Equations & Matrices

Two Variable Systems of Linear Equations

7. Systems of Equations & Matrices

Two Variable Systems of Linear Equations - Online Tutor, Practice Problems & Exam Prep

Two Variable Systems of Linear Equations Practice Problems

125 problems

1PRACTICE PROBLEM

Determine whether the given ordered pairs satisfy the corresponding equations or not.

(i) (3, -2), 3x + 2y = 5

(ii) (3, -2), 7x + 5y = 11

2PRACTICE PROBLEM

A student is conducting an experiment in the laboratory. Determine how many mL of a 12% acid solution that the student must combine with 250 mL of a 24% acid solution to produce a 15% acid solution.

3PRACTICE PROBLEM

Solve the following system of equations. Express x or y in terms of a, if needed. Assume that a is not equal to 0.

4PRACTICE PROBLEM

Use any suitable method to solve the given system:

5PRACTICE PROBLEM

Use any suitable method to solve the given system:

6PRACTICE PROBLEM

Let x stand in for one number and y for the other. Create an equation system using the circumstances provided. Find the numbers by solving the system.

The addition of the first number and the double of the other number is 22.
The difference between these two numbers is -2.

7PRACTICE PROBLEM

Try to solve the following system of equations. Determine if the system has no solution, or infinitely many solutions (in which case express the answer using set notation.)

8PRACTICE PROBLEM

Try to solve the following system of equations. Determine if the system has no solution, or infinitely many solutions (in which case express the answer using set notation.)

9PRACTICE PROBLEM

Try to solve the following system of equations. Determine if the system has no solution, or infinitely many solutions (in which case express the answer using set notation.)

10PRACTICE PROBLEM

Try to solve the following system of equations. Determine if the system has no solution, or infinitely many solutions (in which case express the answer using set notation.)

11PRACTICE PROBLEM

Try to solve the following system of equations. Determine if the system has no solution, or infinitely many solutions (in which case express the answer using set notation.)

12PRACTICE PROBLEM

Try to solve the following system of equations. Determine if the system has no solution, or infinitely many solutions (in which case express the answer using set notation.)

13PRACTICE PROBLEM

Consider the given system of equations and solve using addition method:

14PRACTICE PROBLEM

Consider the given system of equations and solve using the addition method:

15PRACTICE PROBLEM

Consider the given system of equations and solve using addition method:

16PRACTICE PROBLEM

Consider the given system of equations and solve using addition method:

17PRACTICE PROBLEM

Consider the given system of equations and solve using addition method:

18PRACTICE PROBLEM

Consider the given system of equations and solve using the addition method:

19PRACTICE PROBLEM

Implement the substitution method to solve the system:

20PRACTICE PROBLEM

Implement the substitution method to solve the system:

21PRACTICE PROBLEM

Implement the substitution method to solve the system:

22PRACTICE PROBLEM

Implement the substitution method to solve the system:

23PRACTICE PROBLEM

Implement the substitution method to solve the system:

24PRACTICE PROBLEM

Implement the substitution method to solve the system:

25PRACTICE PROBLEM

Implement the substitution method to solve the system:

26PRACTICE PROBLEM

Find out whether the given ordered pair satisfies the system of equations shown.
(1, 3)

27PRACTICE PROBLEM

Find out whether the given ordered pair satisfies the system of equations shown.
(- 5, 1)

28PRACTICE PROBLEM

Find out whether the given ordered pair satisfies the system of equations shown.

(-2, 6)

29PRACTICE PROBLEM

The difference between twice the length and 4 times the width of the lawn shown is 22 meters. Its perimeter is 52 meters. What is the length and width of the lawn?

30PRACTICE PROBLEM

Alice is mixing a 20% concentration of orange juice with an orange juice that has a higher concentration of 40% to have a 250-mL of orange juice mixture with 32% concentration. How many milliliters of each of the orange juice should be used?

31PRACTICE PROBLEM

Solve the following system of equations by using your preferred method. Use the set notation to express their solution sets if the system has no solution or has infinitely many solutions.

32PRACTICE PROBLEM

Solve the following system of equations by using your preferred method. Use the set notation to express their solution sets if the system has no solution or has infinitely many solutions.

33PRACTICE PROBLEM

Using the substitution method, solve the following system of equations.

34PRACTICE PROBLEM

Using the substitution method, solve the following system of equations.

35PRACTICE PROBLEM

Using the substitution method, solve the following system of equations.

36PRACTICE PROBLEM

Using the substitution method, solve the following system of equations.

37PRACTICE PROBLEM

Using the substitution method, solve the following system of equations.

38PRACTICE PROBLEM

Using the substitution method, solve the following system of equations.

39PRACTICE PROBLEM

Using the elimination method, solve the following system of equations.

40PRACTICE PROBLEM

Using the elimination method, solve the following system of equations.

41PRACTICE PROBLEM

Using the elimination method, solve the following system of equations.

42PRACTICE PROBLEM

Using the elimination method, solve the following system of equations.

43PRACTICE PROBLEM

Solve the given system of linear equations using the substitution method.

${"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mfenced open=\"{\" close=\"\"><mtable columnalign=\"left\"><mtr><mtd><mn>4</mn><mi>x</mi><mo> </mo><mo>+</mo><mo> </mo><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mn>16</mn></mtd></mtr><mtr><mtd><mn>6</mn><mi>x</mi><mo> </mo><mo>+</mo><mo> </mo><mn>5</mn><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mn>10</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}$

44PRACTICE PROBLEM

Solve the given system of linear equations using the elimination method.

${"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mfenced open=\"{\" close=\"\"><mtable columnalign=\"left\"><mtr><mtd><mfrac><mrow><mi>x</mi><mo> </mo></mrow><mn>7</mn></mfrac><mo>+</mo><mfrac><mrow><mo> </mo><mi>y</mi><mo> </mo></mrow><mn>14</mn></mfrac><mo>=</mo><mo> </mo><mn>8</mn></mtd></mtr><mtr><mtd><mfrac><mrow><mi>x</mi><mo> </mo></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mrow><mo> </mo><mi>y</mi><mo> </mo></mrow><mn>8</mn></mfrac><mo>=</mo><mo> </mo><mn>14</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}$

45PRACTICE PROBLEM

Solve the given system of linear equations using the elimination method.

${"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mfenced open=\"{\" close=\"\"><mtable columnalign=\"left\"><mtr><mtd><mn>12</mn><mi>x</mi><mo> </mo><mo>+</mo><mo> </mo><mn>8</mn><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>15</mn></mtd></mtr><mtr><mtd><mn>5</mn><mi>x</mi><mo> </mo><mo>+</mo><mo> </mo><mn>3</mn><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}$

46PRACTICE PROBLEM

Solve the following system of equations with two variables and determine if it is an inconsistent system or has infinitely many solutions. Write the solution set in terms of x if it has infinitely many solutions.

47PRACTICE PROBLEM

48PRACTICE PROBLEM

In standard form, write the system of equations associated with the graph shown.

49PRACTICE PROBLEM

In standard form, write the system of equations associated with the graph shown.

50PRACTICE PROBLEM

Solve the following system of equations, where x ≠ 0 and y ≠ 0:

51PRACTICE PROBLEM

Solve the following system of equations, where x ≠ 0 and y ≠ 0:

52PRACTICE PROBLEM

Using the elimination method, solve the following system of equations.

53PRACTICE PROBLEM

Using the elimination method, solve the following system of equations.

54PRACTICE PROBLEM

Find two numbers that sum up to 195 and have a difference of 3.

55PRACTICE PROBLEM

Enumerate the prices of a sack of dog food and a sack of cat food, if, in a pet shop, seven sacks of dog food and six bags of cat food were sold for $627, while three sacks of dog food and eight sacks of cat food were sold for $589 with no discounts applied.

56PRACTICE PROBLEM

Find the value(s) of q at which the system will have infinitely many solutions and no solution:

57PRACTICE PROBLEM

Write an equation of a line that passes through (4, 6) and (-3, -8). Express the equation in slope-intercept form, y = mx + b. Make use of a system of equations in finding the values of m and b.

58PRACTICE PROBLEM

Write an equation of a parabola that passes through (-6, 7), (-3, 4), and (-1, 12). Express the equation in the form y = ax² + bx + c . Make use of a system of equations in finding the values of a, b, and c.

59PRACTICE PROBLEM

A pharmacy sells Azithromycin for $ 10 per tablet and Amoxicillin for $ 15 per tablet. The pharmacy receives an order for 50 tablets, but the customer forgot to specify how many of each item to send. If the pharmacy receives $ 650 for the order, how many tablets of Azithromycin and Amoxicillin should they send?

60PRACTICE PROBLEM

Solve the given system of equations.

61PRACTICE PROBLEM

Solve the system of logarithmic equations given below.

62PRACTICE PROBLEM

Solve the system of logarithmic equations given below.

63PRACTICE PROBLEM

Determine the dimensions of a rectangular wall if its perimeter and area are 40 feet and 64 square feet respectively.

64PRACTICE PROBLEM

Determine the dimensions of a rectangular wall if its perimeter and area are 40 feet and 64 square feet respectively.

65PRACTICE PROBLEM

Let x stand in for one number, and y for the other. Create a system of equations based on the information given below. Find the numbers by solving the system.

The squares of two numbers have a difference of five. Increasing three times the square of the first number by the square of the other number yields 31.

66PRACTICE PROBLEM

Let x stand in for one number, and y for the other. Create a system of equations based on the information given below. Find the numbers by solving the system.

The result of addition of first number and the other number is 14. The product of two numbers is 48.

67PRACTICE PROBLEM

Solve the given system of equations.

68PRACTICE PROBLEM

Solve the given system of equations.

69PRACTICE PROBLEM

Solve the given system of equations.

70PRACTICE PROBLEM

Solve the given system of equations.

71PRACTICE PROBLEM

Solve the given system of equations.

72PRACTICE PROBLEM

Solve the given system of equations.

73PRACTICE PROBLEM

Solve the given system of equations.

74PRACTICE PROBLEM

Solve the system of equations given below using the addition method.

75PRACTICE PROBLEM

Solve the system of equations given below using the addition method.

76PRACTICE PROBLEM

Solve the system of equations given below using the addition method.

77PRACTICE PROBLEM

Solve the system of equations given below using the addition method.

78PRACTICE PROBLEM

Work out the following system of nonlinear equations.

79PRACTICE PROBLEM

Work out the following system of nonlinear equations.

80PRACTICE PROBLEM

Solve the system of equations given below using the addition method.

81PRACTICE PROBLEM

Solve the system of equations given below using the substitution method.

82PRACTICE PROBLEM

Solve the system of equations given below using the substitution method.

83PRACTICE PROBLEM

Solve the system given below using the substitution method.

84PRACTICE PROBLEM

Solve the system given below using the substitution method.

85PRACTICE PROBLEM

Solve the system given below using the substitution method.

86PRACTICE PROBLEM

Solve the system given below using the substitution method.

87PRACTICE PROBLEM

Solve the system given below using the substitution method.

88PRACTICE PROBLEM

Solve the system given below using the substitution method.

89PRACTICE PROBLEM

Solve the system given below using the substitution method.

90PRACTICE PROBLEM

Consider the given rectangle. It has an area of 221 square meters and a perimeter of 60 meters. Find its length and width.

91PRACTICE PROBLEM

Solve the following system of equations using a method of your choice.

x² + y² = 9
x - y = 3

92PRACTICE PROBLEM

Solve the following system of equations using a method of your choice.

y = x² + 1
x + y = 1

93PRACTICE PROBLEM

Solve the following system of equations using a method of your choice.

y = x² + 4x + 4
x + y = 0

94PRACTICE PROBLEM

Out of two solutions of the given nonlinear system, one solution is (0, _):

95PRACTICE PROBLEM

Out of two solutions of the given nonlinear system, one solution is (_, 2):

96PRACTICE PROBLEM

What is the resulting equation when we substitute the solution for y in equation (2) into equation (1) to solve the given nonlinear system using the substitution method?

x² + 2y = 63 (1)

x - y = 0 (2)

97PRACTICE PROBLEM

Find two numbers using a system of nonlinear equations in two variables if they sum up to 64 and yield a product of 295 when multiplied.

98PRACTICE PROBLEM

Using a system of nonlinear equations in two variables, determine the two numbers whose sum of squares is equal to 130 and whose difference of squares is equal to 32.

99PRACTICE PROBLEM

For the indicated intersection points on the graph, check if they are solutions to the nonlinear system of equations given using the substitution method.

7x² = 9y + 9

3y = 7x - 3

100PRACTICE PROBLEM

For the indicated intersection points on the graph, check if they are solutions to the nonlinear system of equations given using the substitution method.

9y = 2x²

x²+ y²= 13