Verifying a Solution

Is $(x, y) = (4, 3)$ a solution to the system: $x + y = 7$ and $2x - y = 5$?

Substitute into the first equation: $x + y = 4 + 3 = 7$ = First equation: $7 = 7$ is TRUE

Substitute into the second equation: $2x - y = 2(4) - 3 = 8 - 3 = 5$ = Second equation: $5 = 5$ is TRUE

Check if BOTH are satisfied: First equation: TRUE, Second equation: TRUE = Both equations are satisfied

Answer: Yes, $(4, 3)$ is a solution because it satisfies both equations.

Finding a Solution by Testing Values

Find the solution to: $x + y = 6$ and $x - y = 2$ by testing values.

Think about what the equations tell us: The sum is 6, the difference is 2. One number is larger than the other. = We need $x > y$

Try $x = 4$, $y = 2$: Sum: $4 + 2 = 6$ (correct!). Difference: $4 - 2 = 2$ (correct!) = Both equations check out

Verify the solution: $x + y = 4 + 2 = 6$, $x - y = 4 - 2 = 2$ = $(4, 2)$ is the solution

Answer: The solution is $x = 4$ and $y = 2$, or $(4, 2)$.

Understanding the Graph

The lines $y = x + 1$ and $y = -x + 5$ intersect at a point. What is that point?

Set the equations equal (both equal $y$): $x + 1 = -x + 5$ = Equation to solve for $x$

Solve for $x$: $x + x = 5 - 1$, so $2x = 4$, giving $x = 2$ = $x = 2$

Find $y$ by substituting back: $y = x + 1 = 2 + 1 = 3$ = $y = 3$

State the intersection point: The point is $(2, 3)$ = This is where both lines cross

Answer: The lines intersect at the point $(2, 3)$.

Real-World System

At a cafe, 2 coffees and 1 tea cost 9 euros. 1 coffee and 2 teas cost 6 euros. What is the price of each drink?

Define variables: Let $c$ = price of coffee, $t$ = price of tea = $c$ and $t$ are our unknowns

Write the equations: $2c + t = 9$ and $c + 2t = 6$ = System of equations formed

Solve using substitution or elimination: From equation 2: $c = 6 - 2t$. Substitute: $2(6 - 2t) + t = 9$ = $12 - 4t + t = 9$, so $-3t = -3$

Find both values: $t = 1$, then $c = 6 - 2(1) = 4$ = Coffee = 4 euros, Tea = 1 euro

Answer: A coffee costs 4 euros and a tea costs 1 euro.

Mistake: Checking only one equation and declaring it a solution

Why: A solution must satisfy ALL equations in the system, not just one.

Correct: Always substitute your answer into every equation to verify it works for all of them.

Mistake: Confusing the solution $(3, 2)$ with $(2, 3)$

Why: In ordered pairs, the first number is $x$ and the second is $y$. The order matters!

Correct: Always write solutions in the correct order: $(x, y)$. Label your work clearly.

Mistake: Thinking parallel lines have a solution

Why: Parallel lines never intersect, so there's no point that lies on both lines.

Correct: If two lines are parallel (same slope, different intercept), the system has no solution.

Shopping Problems

Determining individual prices when you only know combined totals.

If 3 apples and 2 oranges cost 11 euros, and 2 apples and 3 oranges cost 9 euros, you can find the price of each fruit using a system of equations.

Mixture Problems

Combining different substances to achieve a target concentration or quantity.

Mixing a 10% acid solution with a 30% acid solution to create 100 mL of a 22% solution.

Q: What if the lines never intersect?

A: If the lines are parallel (same slope but different y-intercepts), they never meet and the system has no solution. We call this an inconsistent system.

Q: Can a system have more than one solution?

A: If the two equations represent the same line, then every point on that line is a solution. This gives infinitely many solutions. We call this a dependent system.

Q: Why do we need two equations for two unknowns?

A: One equation with two unknowns has infinitely many solutions (a whole line of points). The second equation narrows it down to a specific point where both conditions are met.