Simultaneous Linear Equations

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Graph showing intersecting lines representing simultaneous equations

📊 Part 1: Understanding Simultaneous Equations

1. What does it mean when two linear equations are solved simultaneously?

Finding where the two lines intersect on a graph

Solving each equation separately

Adding the two equations together

Finding the slope of both lines

2. Circle the correct notation for a pair of simultaneous equations:

3x + 4y = 5 and 5x - 6y = 2

3x + 4y = 5 or 5x - 6y = 2

3x + 4y = 5 then 5x - 6y = 2

3x + 4y = 5 plus 5x - 6y = 2

3. Fill in the blanks:

When solving simultaneous equations, we are looking for values of _____ and _____ that satisfy _______ equations at the same time. Graphically, this represents the _____________ point of two lines.

🔄 Part 2: Substitution Method

4. Solve this pair of simultaneous equations using the substitution method:

Equation 1: y = 2x + 1

Equation 2: 3x + y = 11

Solution: x = _______ and y = _______

5. Solve using substitution (you'll need to rearrange first):

Equation 1: 2x + y = 8

Equation 2: x - y = 1

Solution: x = _______ and y = _______

➖ Part 3: Elimination Method

6. Solve this pair using the elimination method:

Equation 1: 3x + 2y = 12

Equation 2: 3x - y = 3

Solution: x = _______ and y = _______

7. Solve using elimination (multiply equations first):

Equation 1: 2x + 3y = 7

Equation 2: 4x - y = 1

Solution: x = _______ and y = _______

📈 Part 4: Graphical Interpretation

8. Plot the following pair of equations on the grid below and find their intersection point:

Equation 1: y = x + 2

Equation 2: y = -x + 4

Intersection point: (_____, _____)

9. Check all statements that are true about simultaneous equations:

The solution represents where two lines cross

Parallel lines have no solution

The same line written differently has infinite solutions

Every pair of equations has exactly one solution

✅ Part 5: Checking Solutions

10. Check if x = 2 and y = 3 is the correct solution for this system:

Equation 1: 2x + y = 7

Equation 2: x - y = -1

Check Equation 1: 2(____) + ____ = ____

Check Equation 2: ____ - ____ = ____

Yes, this is the correct solution

No, this is not the correct solution

11. Match each method with its description:

1. Substitution Method

2. Elimination Method

3. Graphical Method

A. Add or subtract equations to remove a variable

B. Plot both lines and find where they cross

C. Solve for one variable and replace it in the other equation

🌟 Part 6: Real-World Applications

12. A school canteen sells pies for $4 each and sandwiches for $3 each. On Monday, they sold 50 items and made $180. Set up simultaneous equations to find how many pies (p) and sandwiches (s) were sold:

Equation 1 (total items): ________________

Equation 2 (total money): ________________

Answer: _____ pies and _____ sandwiches were sold.

13. Explain in your own words why we need two equations to solve for two unknown variables: