Revision Notes: Linear Equations in Two Variables
- 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.
- 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.
- 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.
- Graphical Representation
A linear equation in two variables represents a straight line on a Cartesian plane. The general process to plot the graph includes:
- Choose any two values for [math] x [/math] to find corresponding [math] y [/math] values.
- Plot these points on the graph.
- Draw a straight line passing through them.
- 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])
- 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.
- Solving Linear Equations in Two Variables
Graphical Method: Plot both equations on the same graph and find the point of intersection.
Substitution Method:
- Solve one equation for one variable in terms of the other.
- Substitute this value into the second equation.
- Solve for the second variable.
Elimination Method:
- Multiply the equations to make the coefficients of one variable equal.
- Subtract or add the equations to eliminate one variable.
- Solve for the other variable.
- 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.
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