Graphical Method And Line Behavior

Connect the algebraic ratios of linear equations to their graphical behavior (intersecting, parallel, coincident).

9 lessons

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Concept

Visualizing Solutions

Explain how two lines behave on a plane.

When we graph a pair of linear equations in two variables, each equation forms a straight line. By looking at how these two lines interact on a coordinate plane, we can immediately understand the solutions to the system.

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Comparing Ratios

Table 3.1 mapping algebraic ratios to graphical outcomes.

Intersecting Lines

Ratio condition

\frac{a _{1}}{a _{2}} \neq = \frac{b _{1}}{b _{2}}

• Exactly one solution

• Unique and consistent pair

Coincident Lines

Ratio condition

\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}

• Infinitely many solutions

• Dependent and consistent pair

Parallel Lines

Ratio condition

\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} \neq = \frac{c _{1}}{c _{2}}

• No solution

• Inconsistent pair

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Worked Example

Example 1: Intersecting Lines

Check graphically if equations are consistent.

Problem. Check graphically whether the following pair of equations is consistent. If so, solve them graphically:

$x + 3y = 6$ $2x - 3y = 12$

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Worked Example

Example 2: Coincident Lines

Check equations for infinitely many solutions.

Problem. Graphically, find whether the following pair of equations has no solution, unique solution, or infinitely many solutions:

$5x - 8y + 1 = 0 \quad \text{--- (1)}$ $3x - \frac{24}{5}y + \frac{3}{5} = 0 \quad \text{--- (2)}$

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reading

Example 3: Champa's Sale

Set up and solve a word problem graphically.

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Let us denote the number of pants by x and the number of skirts by y. The equations formed based on Champa's statements are y = 2x - 2 and y = 4x - 4. We can draw the graphs of these equations by finding two solutions for each line. When plotted on graph paper, the two lines intersect at the point . So, the coordinate values x = 1 and y = provide the required solution for this pair of linear equations. This means the number of pants she purchased is 1 and she bought skirts.

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Worked Example

Example 1: Intersecting Lines

Check graphically if equations are consistent.

Problem. Check graphically whether the following pair of equations is consistent. If so, solve them graphically:

$x + 3y = 6$ $2x - 3y = 12$

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Worked Example

Example 2: Coincident Lines

Check equations for infinitely many solutions.

Problem. Graphically, find whether the following pair of equations has no solution, unique solution, or infinitely many solutions:

$5x - 8y + 1 = 0 \quad \text{--- (1)}$ $3x - \frac{24}{5}y + \frac{3}{5} = 0 \quad \text{--- (2)}$

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