Systems of Equations

The Big Idea

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A “system” is just two (or more) equations with the same variables. The solutions are the points (x, y) where the equations intersect.

We also call these “two equations, two unknowns” problems. 

Two main solving strategies:

1. Substitution → Solve one equation for one variable, plug into the other.
   
   
2. Elimination → Add or subtract the equations to cancel out a variable.
   
   

This video breaks it down using the aforementioned method and using a graphing method:

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Example 1

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Solve the system of equations:
3x + 4y = 25
x + y = 7

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Step 1: Solve the easier equation for one variable.
From x + y = 7 → x = 7 – y

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Step 2: Substitute it into the other equation. 

For every x, substitute in 7-y

3(7 – y) + 4y = 25
21 – 3y + 4y = 25
21 + y = 25

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Step 3: Solve for y.
y = 25 – 21 = 4

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Step 4: Plug y back into an original equation to find x.
x + y = 7 → x + 4 = 7 → x = 3

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Answer: (x, y) = (3, 4)

Teacher tip: Always start with the simpler-looking equation when using substitution.

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Example 2

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Solve the system:
6x + y = 25
2x – y = 7

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Step 1: Notice the y’s are set up to cancel (y and –y). Use elimination.
Add the equations together:
(6x + y) + (2x – y) = 25 + 7 The ys cancel out
8x = 32

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Step 2: Solve for x
x = 32 ÷ 8 = 4

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Step 3: Plug back into one equation
6(4) + y = 25
24 + y = 25
y = 1

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Answer: (x, y) = (4, 1)

Teacher tip: Elimination works beautifully when you see opposite terms (like +y and –y).

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Example 3

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Solve the system:
y = 3x – 2
4x + y = 19

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Step 1: Substitute the expression for y
4x + (3x – 2) = 19
4x + 3x – 2 = 19

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Step 2: Simplify
7x – 2 = 19

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Step 3: Solve for x
7x = 21 → x = 3

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Step 4: Find y by substituting x = 3 into an original equation
y = 3(3) – 2 = 9 – 2 = 7

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Answer: (x, y) = (3, 7)

Teacher tip: When one equation is already solved for y (or x), substitution is the fastest.

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Common Mistakes to Avoid

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1. Dropping a negative → especially when substituting. Write parentheses around substituted values.
   
   
2. Forgetting to plug back in → solving for just one variable isn’t enough; always find both x and y.
   
   
3. Combining unlike terms → don’t mix x’s with numbers before checking.
   
   

This is a longer explanation of these concepts:

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Quick Practice Problems

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Try these:

1. x + y = 10, x – y = 2
   
2. 2x + 3y = 12, x – y = 1
   
3. y = 2x + 1, 3x + y = 19
   
   

Answers:

1. (6, 4) easiest to solve using elimination
   
2. (3, 2) easiest to solve using substitution
   
3. (6, 13) easiest to solve using substitution