Solving Systems of Linear Equations

Generate an educational illustration depicting a person solving a system of linear equations on a chalkboard, with visuals of intersecting lines and graphs in the background.

Mastering Systems of Linear Equations

Test your skills in solving systems of linear equations with our engaging quiz! This quiz features a variety of problems that will challenge your knowledge and enhance your understanding of both substitution and elimination methods.

Participate now and discover how well you can:

Solve equations using substitution
Utilize elimination techniques
Identify solutions graphically

11 Questions3 MinutesCreated by SolvingSage251

A system of linear equations is just what it sounds like - 2 or more equations which can describe lines in the coordinate plane.

If these lines intersect, the point at which they intersect is called the solution to the system!

A system of linear equations is just what it sounds like - 2 or more equations which can describe lines in the coordinate plane.

If these lines intersect, the point at which they intersect is called the solution to the system!

(Image courtesy of Mathnasium)

One method by which we can solve systems of equations is called substitution.

Here's an example:

y - 0.5x = 2

y + 2x = -3

By solving the bottom equation for y, we can get:

y = -2x -3

Then, we can re-write the top equation and replace the y with -2x - 3.

-2x - 3 - 0.5x = 2

Now, we can combine like terms...

-2.5 x = 5

x = -2.

Now that we have a value for x, we can plug that back into the original second equation...

y + 2(5) = -3

y + 10 = -3

y = -13.

And there you have it - substitution!

One method by which we can solve systems of equations is called substitution.

Here's an example:

y - 0.5x = 2

y + 2x = -3

By solving the bottom equation for y, we can get:

y = -2x -3

Then, we can re-write the top equation and replace the y with -2x - 3.

-2x - 3 - 0.5x = 2

Now, we can combine like terms...

-2.5 x = 5

x = -2.

Now that we have a value for x, we can plug that back into the original second equation...

y + 2(5) = -3

y + 10 = -3

y = -13.

And there you have it - substitution!

Now you try!

(questions come from Johns Hopkins CTY Honors Algebra II, KB Gardner instructor)

Now you try!

(questions come from Johns Hopkins CTY Honors Algebra II, KB Gardner instructor)

Solve this system of equations by the substitution method.

x+2y = 5

-x + y = -5

(0,5/2)

(6,11)

(5,0)

No solution

Find the solution of the system by using substitution.

x + 2y = -5

3x + y = -5

(0, −5)

(−1, −2)

(-2, -3/2)

No solution

Solve this system.

4x + 4y = 4

-12x - 12y = -13

Here's a hint: start by factoring out obvious things, but be careful of logical absurdities like 1 = 13/12...

(0,1)

(-3, 4)

(-8, 4)

No Solution

Another method of solving linear equations is by elimination.

In this method, one equation is added to or subtracted from the other to eliminate one of the variables.

For example...

4x + 3y = 2

8x - 3y = 10

Adding these two equations together will eliminate the y variable...

4x + 3y = 2

+ 8x - 3y = 10

12x = 12

x=1.

From here, we can proceed using substitution...

4(1) + 3y = 2

4 + 3y = 2

3y = -2

y = -2/3.

Another method of solving linear equations is by elimination.

In this method, one equation is added to or subtracted from the other to eliminate one of the variables.

For example...

4x + 3y = 2

8x - 3y = 10

Adding these two equations together will eliminate the y variable...

4x + 3y = 2

+ 8x - 3y = 10

12x = 12

x=1.

From here, we can proceed using substitution...

4(1) + 3y = 2

4 + 3y = 2

3y = -2

y = -2/3.

Find the solution of the system by using elimination.

2x-y = -10

x+y = -2

(0, 10)

(-18, 2)

(-4, 2)

No solution

Solve the system by the elimination method.

Hint: find a common denominator first!

(1, 4)

(2, 9/2)

(5, 3)

(4, 11/2)

How did you do?

Apply your knowledge by practicing in this section's homework.

Want to learn more? Try graphing the systems of equations in DESMOS!

How did you do?

Apply your knowledge by practicing in this section's homework.

Want to learn more? Try graphing the systems of equations in DESMOS!

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