Free Algebra
Tutorials!
 
Home
Systems of Linear Equations and Problem Solving
Solving Quadratic Equations
Solve Absolute Value Inequalities
Solving Quadratic Equations
Solving Quadratic Inequalities
Solving Systems of Equations Row Reduction
Solving Systems of Linear Equations by Graphing
Solving Quadratic Equations
Solving Systems of Linear Equations
Solving Linear Equations - Part II
Solving Equations I
Summative Assessment of Problem-solving and Skills Outcomes
Math-Problem Solving:Long Division Face
Solving Linear Equations
Systems of Linear Equations in Two Variables
Solving a System of Linear Equations by Graphing
Ti-89 Solving Simultaneous Equations
Systems of Linear Equations in Three Variables and Matrix Operations
Solving Rational Equations
Solving Quadratic Equations by Factoring
Solving Quadratic Equations
Solving Systems of Linear Equations
Systems of Equations in Two Variables
Solving Quadratic Equations
Solving Exponential and Logarithmic Equations
Solving Systems of Linear Equations
Solving Quadratic Equations
Math Logic & Problem Solving Honors
Solving Quadratic Equations by Factoring
Solving Literal Equations and Formulas
Solving Quadratic Equations by Completing the Square
Solving Exponential and Logarithmic Equations
Solving Equations with Fractions
Solving Equations
Solving Linear Equations
Solving Linear Equations in One Variable
Solving Linear Equations
SOLVING QUADRATIC EQUATIONS USING THE QUADRATIC FORMULA
SOLVING LINEAR EQUATIONS
 
Try the Free Math Solver or Scroll down to Tutorials!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


Systems of Linear Equations in Two Variables

Questions

1. Solve the system by graphing:

3x + y = 2
2x − y = 3

2. Solve the system by graphing:

3. Solve the system by graphing:
y = −2x + 5
3y + 6x = 15

4. Solve the system algebraically, using any method you like:
4x + 3y = 9
3y + 6 = x

5. Solve the system algebraically, using any method you like:
5x + 2y = 5
3x + y = 4

6. Solve the system algebraically, using any method you like:
4x + 2y = 4
3x + y = 4

7. Solve the system algebraically, using any method you like:
9x + 2y = 2
3x + 5y = 5

8. Solve the system algebraically, using any method you like:
6s − 3t = 1
5s + 6t = 15

9. Solve the system algebraically, using any method you like:
0.2x = 0.1y − 1.2
2x − y = 6

Solutions

To solve by sketching, we should use graph paper, or be very careful with the scale as we sketch by hand. Whenever
you are reading a solution off of a graph you need to be very precise! That’s why we prefer algebra to solve systems of
equations.

1. You can sketch this using techniques from previous sections (slope and y-intercept, or getting two points).
Sketch 3x + y = 2:

When x = 0 -> 3(0) + y = 2 -> y = 2, so the ordered pair is (0, 2).
When y = 0 -> 3x + (0) = 2 -> x = 2/3, so the ordered pair is (2/3, 0).

Sketch 2x − y = 3:
When x = 0 -> 2(0) − y = 3 -> y = −3, so the ordered pair is (0,−3).
When y = 0 -> 2x − (0) = 3 -> x = 3/2, so the ordered pair is (3/2, 0).

The solution to the system appears to be (1,−1). Check by substituting into the original equations:
3(1) + (−1) = 2 True
2(1) − (−1) = 3 True

2. You can sketch this using techniques from previous sections (slope and y-intercept, or getting two points).
Sketch y = 1/3x − 2:

When x = 0 -> y =1/3(0) − 2 -> y = −2, so the ordered pair is (0,−2).
When y = 0 -> 0 =1/3x − 2 -> x = 6, so the ordered pair is (6, 0).

Sketch −x + 3y = 9:
When x = 0 -> −(0) + 3y = 9 -> y = 3, so the ordered pair is (0, 3).
When y = 0 -> −x + 3(0) = 9 -> x = −9, so the ordered pair is (−9, 0).

The system has no solution, since the lines are parallel. Check by computing the slope of each line (parallel lines have the same slope).

3. You can sketch this using techniques from previous sections (slope and y-intercept, or getting two points).
Sketch y = −2x + 5:

When x = 0 -> y = −2(0) + 5 -> y = 5, so the ordered pair is (0, 5).
When y = 0 -> 0 = −2x + 5 -> x = 5/2, so the ordered pair is (5/2, 0).

Sketch 3y + 6x = 15:
When x = 0 -> 3y + 6(0) = 15 -> y = 5, so the ordered pair is (0, 5).
When y = 0 -> 3(0) + 6x = 15 -> x = 5/2, so the ordered pair is (5/2, 0).

The system has an infinite number of solutions, since the lines are identical. Check by showing the lines have the same
equation. We can see that second equation is just the first equation multiplied by 3.
y = −2x + 5
3y + 6x = 15

4. Let’s use the substitution method.
From the second equation, we can solve for x = 3y + 6. Substitute this into the first equation:

4x + 3y = 9
4(3y + 6) + 3y = 9 now, solve for y
12y + 24 + 3y = 9
15y = 9 − 24
15y = −15
y = −1

Now, use this value of y in x = 3y + 6 to determine x:
x = 3y + 6
x = 3(−1) + 6
x = 3

The solution to the system is the ordered pair (3,−1). You can check by substituting this back into both original equations.
They should both be true when x = 3 and y = −1.

5. Let’s use the substitution method.
From the second equation, we can solve for y = 4 − 3x. Substitute this into the first equation:

5x + 2y = 5
5x + 2(4 − 3x) = 5 now, solve for x
5x + 8 − 6x = 5
−x = 5 − 8
−x = −3
x = 3

Now, use this value of x in y = 4 − 3x to determine y:
y = 4 − 3x
y = 4 − 3(3)
y = −5

The solution to the system is the ordered pair (3,−5).

6. Let’s use the substitution method.
From the second equation, we can solve for y = 4 − 3x. Substitute this into the first equation:

4x + 2y = 4
4x + 2(4 − 3x) = 4 now, solve for x
4x + 8 − 6x = 4
−2x = 4 − 8
−2x = −4
x = 2

Now, use this value of x in y = 4 − 3x to determine y:
y = 4 − 3x
y = 4 − 3(2)
y = −2

The solution to the system is the ordered pair (2,−2).

7. Let’s use the elimination method.
Multiply the second equation by −3 to make the coefficient of x the same in both equations, but with opposite sign.

9x + 2y = 2
−9x − 15y = −15

Now add the two equations to eliminate the x (since 9x − 9x = 0):
9x + 2y = 2
−9x − 15y = −15

Adding:
2y − 15y = 2 − 15 now solve for y
−13y = −13
y = 1

Now, use this value of y in any of the earlier equations to determine x:
9x + 2y = 2
9x + 2(1) = 2
9x + 2 = 2
9x = 0
x = 0

The solution to the system is the ordered pair (0, 1).

8. Let’s use the elimination method.
Multiply the first equation by 2 to make the coefficient of t the same in both equations, but with opposite sign.
12s − 6t = 2
5s + 6t = 15

Now add the two equations to eliminate the t (since −6t + 6t = 0):
12s − 6t = 2
5s + 6t = 15

Adding:
17s = 17 now solve for s
s = 1

Now, use this value of s in any of the earlier equations to determine t:

The solution to the system is the ordered pair

8. Let’s use the elimination method.
Multiply the first equation by −10 to make the coefficient of x the same in both equations, but with opposite sign.
−2x = −y + 12
2x − y = 6

Now add the two equations to eliminate the x (since −2x + 2x = 0):
−y = −y + 12 + 6
0 = 18

You might think you’ve made a mistake, but you just need to interpret what you’ve found.
Since 0 can never equal 18, there is no solution to the system of equations. Graphically, the two equations represent two
parallel lines.
 

All Right Reserved. Copyright 2005-2024