Pair of Linear Equations in Two Variables - Plot and solve pair of linear equations graphically and interpret intersection outcomes

Solve the pair of linear equations graphically: $x + y = 3$ and $2x + 5y = 12$.

Step 1: Create a table for $x + y = 3$. If $x=0, y=3$ (Point $A(0,3)$). If $x=3, y=0$ (Point $B(3,0)$). Step 2: Create a table for $2x + 5y = 12$. If $x=1, y=2$ (Point $C(1,2)$). If $x=6, y=0$ (Point $D(6,0)$). Step 3: Plot points $A, B$ and draw a line. Plot points $C, D$ and draw a second line. Step 4: Observe the intersection. The lines intersect at point $(1, 2)$. Therefore, $x = 1, y = 2$.

We find two distinct points for each equation by substituting values for $x$ and solving for $y$. By plotting these on a graph, the point where the lines cross $(1, 2)$ represents the unique solution that satisfies both equations.

Determine if the equations $x - 2y = 0$ and $3x + 4y = 20$ are consistent and find the solution graphically.

Step 1: Check coefficients: $a_1=1, b_1=-2, c_1=0$ and $a_2=3, b_2=4, c_2=-20$. Step 2: Calculate ratios: $\frac{a_1}{a_2} = \frac{1}{3}$ and $\frac{b_1}{b_2} = \frac{-2}{4} = -\frac{1}{2}$. Since $\frac{1}{3} \neq -\frac{1}{2}$, the system is consistent with a unique solution. Step 3: Plot $x - 2y = 0$: Points $(0,0), (4,2)$. Step 4: Plot $3x + 4y = 20$: Points $(0,5), (4,2)$. Step 5: The lines intersect at $(4, 2)$. The solution is $x=4, y=2$.

First, the algebraic ratio test confirms that the lines intersect. Graphing reveals the specific coordinate $(4, 2)$ where the lines meet, providing the graphical solution.

1. For $x - 2y = 5$: Points are $(5, 0)$ and $(1, -2)$.
2. For $3x - 6y = 15$: Dividing by 3, we get $x - 2y = 5$, which is the same equation.
3. Since both equations represent the same line, every point on the line is a solution.

Here, $\frac{a_1}{a_2} = \frac{1}{3}$, $\frac{b_1}{b_2} = \frac{-2}{-6} = \frac{1}{3}$, and $\frac{c_1}{c_2} = \frac{-5}{-15} = \frac{1}{3}$. Since all ratios are equal, the lines are coincident.

Step 1: Find at least two solutions for each equation to plot the lines. For $2x + y = 6$:

• If $x = 0$, $y = 6$. Point: $(0, 6)$
• If $y = 0$, $2x = 6 \Rightarrow x = 3$. Point: $(3, 0)$

For $x - 2y = -2$:

• If $x = 0$, $-2y = -2 \Rightarrow y = 1$. Point: $(0, 1)$
• If $y = 0$, $x = -2$. Point: $(-2, 0)$

Step 2: Plot these points on a graph and draw the lines. Step 3: Observe the intersection point. The two lines intersect at the point $(2, 2)$.

Verification: Substitute $x = 2, y = 2$ into both equations: $2(2) + 2 = 4 + 2 = 6$ (True) $2 - 2(2) = 2 - 4 = -2$ (True)

Since the lines intersect at exactly one point, the system is consistent and has a unique solution $(2, 2)$.

The graphical method involves plotting the linear equations as straight lines on the Cartesian plane. The consistency of the system is determined by the intersection. Since $\frac{a_1}{a_2} = \frac{2}{1}$ and $\frac{b_1}{b_2} = \frac{1}{-2}$, then $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, confirming a unique solution.