Finding the Intersection Point

Solve the system by graphing: $$\begin{cases} y = x + 1 \\ y = -x + 5 \end{cases}$$

Graph the first line $y = x + 1$: y-intercept: $(0, 1)$. Slope: $1$ (up 1, right 1). Points: $(0, 1)$, $(1, 2)$, $(2, 3)$ = Line 1 graphed

Graph the second line $y = -x + 5$: y-intercept: $(0, 5)$. Slope: $-1$ (down 1, right 1). Points: $(0, 5)$, $(1, 4)$, $(2, 3)$ = Line 2 graphed

Find where the lines intersect: Both lines pass through $(2, 3)$ = Intersection: $(2, 3)$

Verify the solution: Check $y = x + 1$: $3 = 2 + 1$ \checkmark Check $y = -x + 5$: $3 = -2 + 5$ \checkmark = Solution verified

Answer: The solution is $(2, 3)$, meaning $x = 2$ and $y = 3$.

System with Different Slopes

Solve the system by graphing: $$\begin{cases} y = 2x - 1 \\ y = \frac{1}{2}x + 2 \end{cases}$$

Graph the first line $y = 2x - 1$: y-intercept: $(0, -1)$. Slope: $2$ (up 2, right 1). Points: $(0, -1)$, $(1, 1)$, $(2, 3)$ = Line 1 graphed

Graph the second line $y = \frac{1}{2}x + 2$: y-intercept: $(0, 2)$. Slope: $\frac{1}{2}$ (up 1, right 2). Points: $(0, 2)$, $(2, 3)$, $(4, 4)$ = Line 2 graphed

Find the intersection point: Both lines pass through $(2, 3)$ = Intersection: $(2, 3)$

Verify the solution: Check $y = 2x - 1$: $3 = 2(2) - 1 = 3$ \checkmark Check $y = \frac{1}{2}x + 2$: $3 = \frac{1}{2}(2) + 2 = 3$ \checkmark = Solution verified

Answer: The solution is $(2, 3)$.

Parallel Lines (No Solution)

Solve the system by graphing: $$\begin{cases} y = 2x + 1 \\ y = 2x - 3 \end{cases}$$

Graph the first line $y = 2x + 1$: y-intercept: $(0, 1)$. Slope: $2$ = Line 1 graphed

Graph the second line $y = 2x - 3$: y-intercept: $(0, -3)$. Slope: $2$ = Line 2 graphed

Look for intersection: Both lines have slope $2$ but different y-intercepts. They are parallel! = Lines never intersect

Determine the solution: Parallel lines never meet, so there is no point that satisfies both equations = No solution

Answer: **No solution**. The lines are parallel and never intersect. This is called an **inconsistent system**.

Same Line (Infinite Solutions)

Solve the system by graphing: $$\begin{cases} y = 3x - 2 \\ 2y = 6x - 4 \end{cases}$$

Graph the first line $y = 3x - 2$: y-intercept: $(0, -2)$. Slope: $3$ = Line 1 graphed

Simplify and graph the second equation: Divide by 2: $y = 3x - 2$. Same equation! = Same line!

Analyze the result: Both equations represent the exact same line = Lines coincide

Determine the solution: Every point on the line satisfies both equations = Infinitely many solutions

Answer: **Infinitely many solutions**. The equations represent the same line. This is called a **dependent system**.

Mistake: Reading the intersection point incorrectly from the graph

Why: Graphs can be imprecise, especially when the intersection has non-integer coordinates.

Correct: Always verify your answer by substituting back into BOTH original equations.

Mistake: Forgetting to check the solution in both equations

Why: A point might satisfy one equation but not the other.

Correct: The solution must work in BOTH equations. Check each one separately.

Mistake: Assuming parallel lines have a solution

Why: Lines with the same slope but different y-intercepts never intersect.

Correct: If slopes are equal and y-intercepts differ, there is NO solution (inconsistent system).

Mistake: Graphing errors from incorrect slope calculation

Why: Rise over run must be applied correctly. A slope of $-2$ means down 2, right 1.

Correct: For positive slope: go up and right. For negative slope: go down and right.

Break-Even Analysis

Businesses use systems to find when revenue equals costs.

A company's costs are $y = 500 + 2x$ (500 euros fixed + 2 euros per item) and revenue is $y = 5x$ (5 euros per item sold). The break-even point is where these lines intersect.

Meeting Point Problem

Determine when and where two moving objects will meet.

Car A starts 10 km ahead and travels at 60 km/h. Car B starts at the origin and travels at 80 km/h. Their positions are $y = 60x + 10$ and $y = 80x$. Where do they meet?

Supply and Demand

Economists find equilibrium where supply equals demand.

Supply: $y = 2x + 10$ (price increases as quantity increases). Demand: $y = -x + 40$ (price decreases as quantity increases). The equilibrium is where these curves intersect.

Q: What if the intersection point has decimal or fraction coordinates?

A: Graphing can be imprecise for non-integer solutions. In such cases, algebraic methods (substitution or elimination) give exact answers. However, you can estimate from the graph and verify algebraically.

Q: How do I know if lines are parallel without graphing?

A: Compare the slopes. If both equations have the same slope (coefficient of $x$ in $y = mx + b$ form) but different y-intercepts, the lines are parallel.

Q: Can a system have exactly two solutions?

A: No. Two straight lines can intersect at most once. A system of linear equations has either 0, 1, or infinitely many solutions.