Revision Notes: Linear Equations in Two Variables

1. Definition of a Linear Equation in Two Variables

A linear equation in two variables is an equation of the form:
[math] ax + by + c = 0 [/math]
where:

[math] a, b, [/math] and [math] c [/math] are real numbers,

[math] a \neq 0 [/math] or [math] b \neq 0 [/math],

[math] x [/math] and [math] y [/math] are variables.

2. Standard Form of Linear Equations

The standard form is written as:
[math] ax + by = c [/math]
where [math] c = -c [/math] from the general form.

3. Solution of a Linear Equation

A solution of a linear equation in two variables is an ordered pair [math] (x, y) [/math] that satisfies the equation.

4. Graphical Representation

A linear equation in two variables represents a straight line on a Cartesian plane. The general process to plot the graph includes:

1. Choose any two values for [math] x [/math] to find corresponding [math] y [/math] values.
2. Plot these points on the graph.
3. Draw a straight line passing through them.
4. Intercepts

X-intercept: The point where the line intersects the x-axis, obtained by setting [math] y = 0 [/math].
[math] x = -\frac{c}{a} [/math] (if [math] a \neq 0 [/math])

Y-intercept: The point where the line intersects the y-axis, obtained by setting [math] x = 0 [/math].
[math] y = -\frac{c}{b} [/math] (if [math] b \neq 0 [/math])

6. Types of Solutions for a Pair of Linear Equations

Consider two linear equations:

[math] a_1x + b_1y + c_1 = 0 [/math]

[math] a_2x + b_2y + c_2 = 0 [/math]

The relationship between the coefficients determines the nature of solutions:

Unique solution: If [math] \frac{a_1}{a_2} \neq \frac{b_1}{b_2} [/math], the lines intersect at one point.

Infinite solutions: If [math] \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} [/math], the lines coincide.

No solution: If [math] \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} [/math], the lines are parallel.

7. Solving Linear Equations in Two Variables

Graphical Method: Plot both equations on the same graph and find the point of intersection.

Substitution Method:

1. Solve one equation for one variable in terms of the other.
2. Substitute this value into the second equation.
3. Solve for the second variable.

Elimination Method:

1. Multiply the equations to make the coefficients of one variable equal.
2. Subtract or add the equations to eliminate one variable.
3. Solve for the other variable.
4. Applications of Linear Equations

Linear equations in two variables are used to solve real-life problems involving relationships between two quantities, such as:

Cost and quantity,

Distance and time,

Supply and demand.

This concludes the revision notes for Linear Equations in Two Variables.

Linear Equations in Two Variables

1. Definition

A linear equation in two variables is an equation of the form:

[math] ax + by + c = 0 [/math]

where:

• a, b, and c are real numbers,
• a ≠ 0 or b ≠ 0,
• x and y are variables.

2. Standard Form

The standard form of a linear equation is:

[math] ax + by = c [/math]

3. Solution of a Linear Equation

The solution is an ordered pair [math](x, y)[/math] that satisfies the equation.

4. Graphical Representation

A linear equation represents a straight line on a Cartesian plane. To plot:

• Choose two values for x and find corresponding y values.
• Plot these points and draw a straight line through them.

5. Intercepts

• X-intercept: Set y = 0 to find [math]x = -\frac{c}{a}[/math].
• Y-intercept: Set x = 0 to find [math]y = -\frac{c}{b}[/math].

6. Types of Solutions for a Pair of Linear Equations

Consider the equations:

[math] a_1x + b_1y + c_1 = 0 [/math]

[math] a_2x + b_2y + c_2 = 0 [/math]

Based on the coefficients:

• Unique Solution: [math]\frac{a_1}{a_2} \neq \frac{b_1}{b_2}[/math].
• Infinite Solutions: [math]\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}[/math].
• No Solution: [math]\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}[/math].

7. Solving Methods

• Graphical Method: Plot both equations and find the intersection point.
• Substitution Method: Solve one equation for one variable and substitute into the other.
• Elimination Method: Manipulate equations to eliminate one variable and solve.

8. Applications

Used in real-life scenarios like cost and quantity, distance and time, and supply and demand problems.