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Introduction to Systems of Equations
Introduction to Systems of Equations
Imagine you're at a carnival, and you want to figure out the cost of a ticket and a snack. You know two things:
If you buy 2 tickets and 3 snacks, it costs $20.
If you buy 1 ticket and 2 snacks, it costs $12.
Make "Let Statements"
Make "Let Statements"
Again, if you buy 2 tickets and 3 snacks, it costs $20. If you buy 1 ticket and 2 snacks, it costs $12.
These two pieces of information are like clues in a puzzle! A system of equations is just a way of writing these clues using math.
Start by writing Let Statements about what you know:
Let t = the cost of a ticket.
Let s = the cost of a snack.
Turn English into mathematical sentences!
Turn English into mathematical sentences!
"If you buy 2 tickets and 3 snacks, it costs $20." This equals: 2t + 3s = 20
"If you buy 1 ticket and 2 snacks, it costs $12." This equals: t + 2s = 12
Let's use Substitution!
Let's use Substitution!
Now we need to discover the value of t and s that make both equations true at the same time. We need to make one of the letters go away. Let's use substitution to do this!
First, rearrange: t + 2s = 12!
It will become: t = 12 - 2s
Since t = 12 - 2s, we can substitute 12 - 2s for t.
Write our first equation again:
2t + 3s = 20
Substitute 12 - 2s for t
2(12 - 2s) + 3s = 20
Distribute: 2(12 - 2s) + 3s = 20
Simplify: 24 - 4s + 3s = 20
24 - s = 20
-20 + s -20 + s
4 = s
4 snacks were purchased!
Determine the value of the other variable, t.
Determine the value of the other variable, t.
Since we know that 4 snacks were purchased, we can use either equation to determine how many tickets were purchased.
2t + 3s = 20
2t + 3 * 4 = 20
2t + 12 = 20
-12 -12
2t = 8
t = 4
4 tickets were purchased!
The answer to both equations is: (4, 4)
The answer to both equations is: (4, 4)
Check your answers!
Check your answers!
Check your work!
Check your work!
2t + 3s = 20
2 * 4 + 3 * 4 = 20
8 + 12 = 20 ✔️
Check your work!
Check your work!
t + 2s = 12
4 + 2 * 4 = 12
4 + 8 = 12 ✔️
This graph of both lines shows the point (4, 4) that makes both equations true.
Three Methods to Solve Systems of Equations:
Three Methods to Solve Systems of Equations:
Equal Values
Equal Values
y = 2x + 7
y = -3x + 12
2x + 7 = -3x + 12
5x = 5
x = 1
y = 9
Substitution
Substitution
2x + 5y = 19
y = -3x + 9
2x + 5(-3x + 9) = 19
2x + -15x + 45 = 19
26 = 13x
x = 2
y = 3
Elimination
Elimination
3y = 2x - 3
y = -2x + 15
3y = 2x - 3
y = -2x + 15
4y = 12
y = 3
x = 6
Why Systems of Equations Matter
Why Systems of Equations Matter
Systems of equations are used all the time in real life:
Systems of equations are used all the time in real life:
- Planning budgets or splitting costs.
- Designing and programming video games.
- Solving mysteries like how many apples and oranges someone bought if you only know the total price and number of items.
By learning how to solve these systems, you'll sharpen your problem-solving skills and learn how to approach tricky situations step-by-step.
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