Systems of Linear Equations

What happens when two lines meet? A system of linear equations helps us find that intersection point! Systems appear everywhere in real life - from business decisions to science problems. Let's explore how to solve them!

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What Is a System of Linear Equations?

A system of linear equations is a set of two or more linear equations with the same variables.

Example system: y = 2x + 1 y = -x + 4

The solution is the point(s) where the lines intersect - the (x, y) values that satisfy BOTH equations simultaneously.

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Types of Solutions

One Solution:

• Lines intersect at exactly one point
• Different slopes
• Most common case

Example: y = 2x + 1 y = -x + 4

These intersect at (1, 3)

No Solution:

• Lines are parallel (never intersect)
• Same slope, different y-intercepts
• Inconsistent system

Example: y = 3x + 2 y = 3x - 5

Same slope (3), different intercepts → parallel!

Infinitely Many Solutions:

• Lines are identical (same line!)
• Same slope, same y-intercept
• Dependent system

Example: y = 2x + 3 2y = 4x + 6 (simplifies to y = 2x + 3)

Same line → infinite solutions!

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Method 1: Graphing

Steps:

1. Graph both equations on same coordinate plane
2. Find the intersection point
3. Check the solution in both equations

Example: Solve by graphing: y = x + 1 y = -2x + 4

Solution:

Graph y = x + 1:

• Y-intercept: (0, 1)
• Slope: 1 (up 1, right 1)
• Points: (0, 1), (1, 2), (2, 3)

Graph y = -2x + 4:

• Y-intercept: (0, 4)
• Slope: -2 (down 2, right 1)
• Points: (0, 4), (1, 2), (2, 0)

Intersection: Lines cross at (1, 2)

Check: y = x + 1: 2 = 1 + 1 ✓ y = -2x + 4: 2 = -2(1) + 4 = 2 ✓

Answer: (1, 2)

Limitations of graphing:

• Not precise for non-integer solutions
• Time-consuming
• Hard to see if lines are exactly parallel

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Method 2: Substitution

Best when one equation is already solved for a variable

Steps:

1. Solve one equation for one variable (if not already)
2. Substitute that expression into the other equation
3. Solve for the remaining variable
4. Substitute back to find the other variable
5. Check solution

Example 1: Solve: y = 3x - 2 2x + y = 8

Solution:

Step 1: First equation already solved for y ✓

Step 2: Substitute y = 3x - 2 into second equation 2x + (3x - 2) = 8

Step 3: Solve for x 2x + 3x - 2 = 8 5x - 2 = 8 5x = 10 x = 2

Step 4: Substitute x = 2 into y = 3x - 2 y = 3(2) - 2 = 6 - 2 = 4

Step 5: Check in both equations y = 3x - 2: 4 = 3(2) - 2 = 4 ✓ 2x + y = 8: 2(2) + 4 = 8 ✓

Answer: (2, 4)

Example 2: Solve: x + 2y = 7 3x - y = 5

Solution:

Step 1: Solve first equation for x x = 7 - 2y

Step 2: Substitute into second equation 3(7 - 2y) - y = 5

Step 3: Solve for y 21 - 6y - y = 5 21 - 7y = 5 -7y = -16 y = 16/7

Step 4: Substitute back x = 7 - 2(16/7) = 7 - 32/7 = 49/7 - 32/7 = 17/7

Answer: (17/7, 16/7) or approximately (2.43, 2.29)

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Method 3: Elimination (Addition)

Best when coefficients line up nicely

Steps:

1. Arrange equations in standard form (Ax + By = C)
2. Multiply one or both equations to make coefficients of one variable opposites
3. Add equations to eliminate that variable
4. Solve for remaining variable
5. Substitute back to find other variable
6. Check solution

Example 1: Solve: 2x + 3y = 12 x - 3y = -3

Solution:

Step 1: Already in standard form ✓

Step 2: The y-coefficients are already opposites (3 and -3) ✓

Step 3: Add equations 2x + 3y = 12

• (x - 3y = -3)

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3x + 0y = 9 3x = 9 x = 3

Step 4: Substitute x = 3 into first equation 2(3) + 3y = 12 6 + 3y = 12 3y = 6 y = 2

Step 5: Check 2x + 3y = 12: 2(3) + 3(2) = 6 + 6 = 12 ✓ x - 3y = -3: 3 - 3(2) = 3 - 6 = -3 ✓

Answer: (3, 2)

Example 2: Solve (requires multiplication): 3x + 2y = 16 5x - y = 7

Solution:

Step 1: Standard form ✓

Step 2: Multiply second equation by 2 to make y-coefficients opposites 3x + 2y = 16 2(5x - y = 7) → 10x - 2y = 14

Step 3: Add 3x + 2y = 16

• (10x - 2y = 14)

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13x = 30 x = 30/13

Step 4: Substitute into 5x - y = 7 5(30/13) - y = 7 150/13 - y = 7 -y = 7 - 150/13 = 91/13 - 150/13 = -59/13 y = 59/13

Answer: (30/13, 59/13) or approximately (2.31, 4.54)

Example 3: Solve (both need multiplication): 2x + 3y = 13 3x + 4y = 18

Solution:

Step 1: Multiply first by 3, second by -2 to eliminate x 6x + 9y = 39 -6x - 8y = -36

Step 2: Add y = 3

Step 3: Substitute y = 3 into 2x + 3y = 13 2x + 3(3) = 13 2x + 9 = 13 2x = 4 x = 2

Answer: (2, 3)

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Identifying No Solution or Infinite Solutions

No Solution (Parallel Lines):

Example: 2x + y = 5 2x + y = 8

When you try to solve: 2x + y = 5 -(2x + y = 8)

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0 = -3 (FALSE!)

Result: No solution (inconsistent)

Infinite Solutions (Same Line):

Example: x + 2y = 6 2x + 4y = 12

When you try to solve: 2(x + 2y = 6) → 2x + 4y = 12 -(2x + 4y = 12)

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0 = 0 (TRUE!)

Result: Infinitely many solutions (dependent)

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Real-World Applications

Business Problem:

Problem: Movie tickets cost $8 for adults and$5 for children. A group buys 20 tickets for $115. How many of each?

Solution:

Let a = adult tickets, c = child tickets

Equation 1: a + c = 20 (total tickets) Equation 2: 8a + 5c = 115 (total cost)

Solve by substitution: From equation 1: a = 20 - c

Substitute into equation 2: 8(20 - c) + 5c = 115 160 - 8c + 5c = 115 160 - 3c = 115 -3c = -45 c = 15

Then: a = 20 - 15 = 5

Answer: 5 adult tickets, 15 child tickets

Mixture Problem:

Problem: How many liters of 20% acid solution and 50% acid solution should be mixed to get 30 liters of 35% acid solution?

Solution:

Let x = liters of 20% solution, y = liters of 50% solution

Equation 1: x + y = 30 (total volume) Equation 2: 0.20x + 0.50y = 0.35(30) = 10.5 (acid amount)

Solve: From equation 1: y = 30 - x

Substitute: 0.20x + 0.50(30 - x) = 10.5 0.20x + 15 - 0.50x = 10.5 -0.30x = -4.5 x = 15

Then: y = 30 - 15 = 15

Answer: 15 L of 20% solution, 15 L of 50% solution

Break-Even Analysis:

Problem: Company A charges $50 setup +$10 per item. Company B charges $20 per item with no setup fee. When do they cost the same?

Solution:

Cost A: y = 10x + 50 Cost B: y = 20x

Set equal: 10x + 50 = 20x 50 = 10x x = 5

Answer: At 5 items, both cost $100 (break-even point)

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Choosing the Best Method

Use Graphing when:

• Quick estimate needed
• Visual representation helpful
• Solutions are integers

Use Substitution when:

• One variable is already isolated
• Equations like y = mx + b
• Fractions can be avoided

Use Elimination when:

• Both equations in standard form
• Coefficients are easy to match
• No variable is isolated

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Common Mistakes to Avoid

❌ Mistake 1: Forgetting to multiply ALL terms

• Wrong: 2(3x + y = 5) → 6x + y = 10
• Right: 2(3x + y = 5) → 6x + 2y = 10

❌ Mistake 2: Sign errors when subtracting

• Be careful with negatives!
• Subtracting an equation = multiply by -1 then add

❌ Mistake 3: Not checking the solution

• Always verify in BOTH original equations

❌ Mistake 4: Stopping after finding one variable

• You need both x AND y!

❌ Mistake 5: Mixing up coordinates

• Solution is (x, y), not (y, x)

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Problem-Solving Strategy

Step 1: Define variables clearly

Step 2: Write two equations from the problem

Step 3: Choose best solution method

• Graphing, substitution, or elimination?

Step 4: Solve the system carefully

Step 5: Check the solution in both equations

Step 6: Answer in context

• Include units!
• Answer the question asked

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Quick Reference

Three Solution Methods:

Graphing:

• Graph both lines
• Find intersection point

Substitution:

• Solve for one variable
• Substitute into other equation

Elimination:

• Make coefficients opposites
• Add equations to eliminate variable

Types of Solutions:

• One solution: Lines intersect (different slopes)
• No solution: Parallel lines (same slope, different intercepts)
• Infinite solutions: Same line (identical equations)

Checking for Special Cases:

• Eliminate variable → 0 = 0? Infinite solutions
• Eliminate variable → 0 = (non-zero)? No solution

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Practice Tips

Tip 1: Organize your work

• Write equations clearly
• Show each step
• Keep = signs aligned

Tip 2: Check for easy elimination first

• Look for coefficients that are already opposites
• Or same coefficients (subtract equations)

Tip 3: Use fractions carefully

• Can multiply to clear denominators
• Or use calculator for decimal approximations

Tip 4: Make a plan before starting

• Which method seems easiest?
• What's the first step?

Tip 5: Verify your answer

• Plug back into BOTH original equations
• Does it make sense in context?

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Summary

Systems of linear equations have two or more equations to solve together:

Solution types:

• One solution (intersection point)
• No solution (parallel lines)
• Infinitely many solutions (same line)

Solution methods:

• Graphing: Visual, good for estimates
• Substitution: Best when variable isolated
• Elimination: Best for standard form

Real-world applications:

• Business and money problems
• Mixture problems
• Break-even analysis
• Comparison problems

Key skill: Translate word problems into two equations, then solve using the most efficient method!

Systems of equations are essential tools for solving complex real-world problems in business, science, engineering, and everyday life!