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1. Word Knowledge
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2.1 Foundational math
2.2 Pre-algebra
2.3 Algebra I
2.4 Geometry basics
2.5 Algebra II
2.5.1 FOIL and multiplying polynomials
2.5.2 Systems of equations
2.5.3 Functions, complex numbers, and consecutive integers
2.5.4 Mean, median, mode, range, and factorials
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2.5.2 Systems of equations
Achievable ASVAB
2. Mathematics Knowledge
2.5. Algebra II
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Systems of equations

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Definitions
System of equations
A set of multiple equations with multiple variables

A system of equations is a set of two or more equations with multiple variables you need to identify. The most common type of system problem on the ASVAB is a two-variable system with two equations.

For example, let’s say we have the system:

2x+5y=−1

−3x−10y=4

We want to solve for the values of both x and y.

The answer to this system is x=2 and y=−1, but how did we get there?

There are multiple ways to solve a systems problem, but let’s look at the main two: elimination and substitution.

Elimination method

Definitions
Elimination
Cancelling out a variable in a system of equations

The elimination method is a way to solve a system by getting rid of one variable so you can solve for the other. We do this by lining up the equations and adding or subtracting them until one variable cancels out.

Signs in elimination: Once the coefficients match, look at their signs. If they have the same sign, subtract the second equation from the first. If they have opposite signs, add the two equations.

Example:

To use elimination, we line up the equations vertically and then either add or subtract them to eliminate one of the variables. So let’s first line them up.

2x+5y=−1

−3x−10y=4

Our goal is to get the coefficients for either x or y to match so we can eliminate that variable. For example, we can eliminate y by getting both equations to have a +/−10y term.

In this case, we can get the top equation with 5y to become 10y by multiplying by 2. Be cautious! Whenever we multiply one value in an equation, we must multiply each value in that equation by the same amount.

2(2x+5y=−1)→(4x+10y=−2)

Now line them up:

4x+10y=−2

−3x−10y=4

The y terms now have the same value with opposite signs. Because their signs are opposite, we’ll add both equations to eliminate y.

4x+10y​=−2

−3x−10y​=4

4x−3x=−2+4

x=2

Now we know the value of one variable.

To find y, plug x=2 into one of the original equations:

2x+5y=−1

2(2)+5y=−1

4+5y=−1

5y=−5

y=−1

We’ve found both values: x=2 and y=−1.

Knowledge check:

Solve using elimination:

x+yx−y​=6=2​

(spoiler)

x=4,y=2

Substitution method

Definitions
Substitution
Isolating and replacing a variable in a system of equations

As an alternative to elimination, we can use substitution.

To use substitution, we solve one equation for one variable and then plug that expression into the other equation.

Example:

2x+4y=14

5x−2y=−25

First, isolate x in the first equation:

2x+4y=14

2x=−4y+14

x=−2y+7

Now replace x in the second equation:

5x−2y=−25

5(−2y+7)−2y=−25

Now we have one variable:

−10y+35−2y=−25

−12y=−60

y=5

Now plug y=5 back in:

x=−2y+7

x=−2(5)+7

x=−10+7

x=−3

So the solution is x=−3 and y=5.

Knowledge check:

Solve using substitution:

x2x+y​=y+1=8​

(spoiler)

x=3,y=2

When to use each method

Some people prefer one method over the other, while others choose based on the problem.

Here are some quick tips:

  • If one variable is easy to isolate, use substitution
  • If coefficients already match (or are easy to match), use elimination
  • If only one equation needs multiplying, elimination is usually easier
  • If both equations need multiplying, substitution may be simpler

Knowledge check:

Which method would be easiest to use for this problem?

x3x+y​=2y−3=10​

(spoiler)

Substitution, because one variable is already isolated.

Systems of equations basics

  • System = two or more equations with multiple unknowns
  • ASVAB focus: two-variable, two-equation systems
  • Two main solving methods: elimination and substitution

Elimination method

  • Goal: match coefficients of one variable, then add/subtract equations to cancel it
  • Rule: same signs → subtract; opposite signs → add
  • Multiply every term in an equation by the same factor to match coefficients
  • After eliminating one variable, solve for it, then substitute back into an original equation to find the other variable

Substitution method

  • Isolate one variable in one equation (solve for x or y)
  • Plug that expression into the other equation to reduce to one variable
  • Solve for remaining variable, then back-substitute to find the first variable

Choosing a method

  • Variable already isolated (or easy to isolate) → use substitution
  • Coefficients match or are easy to match → use elimination
  • Only one equation needs multiplying → elimination is simpler
  • Both equations need multiplying → substitution may be easier

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Next  | 2.5.3 Functions, complex numbers, and consecutive integers
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Systems of equations

Definitions
System of equations
A set of multiple equations with multiple variables

A system of equations is a set of two or more equations with multiple variables you need to identify. The most common type of system problem on the ASVAB is a two-variable system with two equations.

For example, let’s say we have the system:

2x+5y=−1

−3x−10y=4

We want to solve for the values of both x and y.

The answer to this system is x=2 and y=−1, but how did we get there?

There are multiple ways to solve a systems problem, but let’s look at the main two: elimination and substitution.

Elimination method

Definitions
Elimination
Cancelling out a variable in a system of equations

The elimination method is a way to solve a system by getting rid of one variable so you can solve for the other. We do this by lining up the equations and adding or subtracting them until one variable cancels out.

Signs in elimination: Once the coefficients match, look at their signs. If they have the same sign, subtract the second equation from the first. If they have opposite signs, add the two equations.

Example:

To use elimination, we line up the equations vertically and then either add or subtract them to eliminate one of the variables. So let’s first line them up.

2x+5y=−1

−3x−10y=4

Our goal is to get the coefficients for either x or y to match so we can eliminate that variable. For example, we can eliminate y by getting both equations to have a +/−10y term.

In this case, we can get the top equation with 5y to become 10y by multiplying by 2. Be cautious! Whenever we multiply one value in an equation, we must multiply each value in that equation by the same amount.

2(2x+5y=−1)→(4x+10y=−2)

Now line them up:

4x+10y=−2

−3x−10y=4

The y terms now have the same value with opposite signs. Because their signs are opposite, we’ll add both equations to eliminate y.

4x+10y​=−2

−3x−10y​=4

4x−3x=−2+4

x=2

Now we know the value of one variable.

To find y, plug x=2 into one of the original equations:

2x+5y=−1

2(2)+5y=−1

4+5y=−1

5y=−5

y=−1

We’ve found both values: x=2 and y=−1.

Knowledge check:

Solve using elimination:

x+yx−y​=6=2​

(spoiler)

x=4,y=2

Substitution method

Definitions
Substitution
Isolating and replacing a variable in a system of equations

As an alternative to elimination, we can use substitution.

To use substitution, we solve one equation for one variable and then plug that expression into the other equation.

Example:

2x+4y=14

5x−2y=−25

First, isolate x in the first equation:

2x+4y=14

2x=−4y+14

x=−2y+7

Now replace x in the second equation:

5x−2y=−25

5(−2y+7)−2y=−25

Now we have one variable:

−10y+35−2y=−25

−12y=−60

y=5

Now plug y=5 back in:

x=−2y+7

x=−2(5)+7

x=−10+7

x=−3

So the solution is x=−3 and y=5.

Knowledge check:

Solve using substitution:

x2x+y​=y+1=8​

(spoiler)

x=3,y=2

When to use each method

Some people prefer one method over the other, while others choose based on the problem.

Here are some quick tips:

  • If one variable is easy to isolate, use substitution
  • If coefficients already match (or are easy to match), use elimination
  • If only one equation needs multiplying, elimination is usually easier
  • If both equations need multiplying, substitution may be simpler

Knowledge check:

Which method would be easiest to use for this problem?

x3x+y​=2y−3=10​

(spoiler)

Substitution, because one variable is already isolated.

Key points

Systems of equations basics

  • System = two or more equations with multiple unknowns
  • ASVAB focus: two-variable, two-equation systems
  • Two main solving methods: elimination and substitution

Elimination method

  • Goal: match coefficients of one variable, then add/subtract equations to cancel it
  • Rule: same signs → subtract; opposite signs → add
  • Multiply every term in an equation by the same factor to match coefficients
  • After eliminating one variable, solve for it, then substitute back into an original equation to find the other variable

Substitution method

  • Isolate one variable in one equation (solve for x or y)
  • Plug that expression into the other equation to reduce to one variable
  • Solve for remaining variable, then back-substitute to find the first variable

Choosing a method

  • Variable already isolated (or easy to isolate) → use substitution
  • Coefficients match or are easy to match → use elimination
  • Only one equation needs multiplying → elimination is simpler
  • Both equations need multiplying → substitution may be easier

More from Algebra II

  • FOIL and multiplying polynomials
  • Functions, complex numbers, and consecutive integers
  • Mean, median, mode, range, and factorials