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The focus of this section is on linear equations represented as ax + b = 0, explaining how they can be expressed as linear equations in two variables. Examples are provided to demonstrate the conversion and solutions of such equations.
In this section, we explore linear equations that take the specific form of ax + b = 0. Such equations exemplify linear equations in two variables since they can be rearranged and rewritten in the standard form of linear equations, which is ax + by + c = 0. Here, the variable associated with y can be treated as zero, allowing us to express any linear equation in one variable as a special instance of a linear equation in two variables. The significance of this lays in recognizing the broader applications of linear equations and their solutions in various scenarios, such as graphical representation and real-world problem-solving.
Linear Equation: An equation that can be plotted on a graph as a straight line.
Solution: A point that satisfies the equation when substituted.
For every x, a y can be found, solutions abound, all around!
Imagine x and y as friends running a race, each time x runs a step, y matches their pace.
Remember A.B.C: Always balance constants.
{'example': 'Rewrite the following in the form ax + by + c = 0: 2x + 3y = 9.35.', 'solution': '2x + 3y - 9.35 = 0.'}
{'example': 'Express 4 - 3x = 0 in two variables.', 'solution': '-3x + 0y + 4 = 0.'}
Term: Linear Equation
Definition: An equation of the form ax + b = 0 where a and b are constants, and a ≠ 0.
An equation of the form ax + b = 0 where a and b are constants, and a ≠ 0.
Term: Two Variables
Definition: Variables in mathematical equations that represent two unknown quantities, typically expressed with x and y.
Variables in mathematical equations that represent two unknown quantities, typically expressed with x and y.
Term: Solution
Definition: A value or set of values that satisfy the equation.
A value or set of values that satisfy the equation.