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Linear Equations in Two Variables A linear equation is an equation of the form. Notice that this equation has variables: and . Consider the equation: - The - The - The term is term is term is This equation has an infinite number of solutions and these solutions are expressed as Ordered Pairs written in the form ( Order is very important, hence the name Ordered Pairs. The always first and the is always second. ). is To find some solutions to the equation We make a chart: To fill out the “ ” column, we substitute for and solve for . If , we have ( ) 1 So we can fill out the chart As an ordered pair, we write this solution ( when , . ) indicating that Consider the equation: To find ordered pairs that satisfy this equation we set up a chart: Fill in the corresponding values NOTE: We can choose any value for and find its corresponding value, but it is helpful to choose small, whole numbers to make the calculations a bit easier. 2 Example 1: Find three solutions to the equation Plotting Solutions Consider the equation The solution to this single variable equation is . We can plot this solution on a number line: 𝟑 𝟐 𝟏 𝟎 𝟏 𝟐 𝟑 Since linear equations have two variables, we require two number lines to plot solution points. 𝒚-axis 𝒙-axis 3 Cartesian Coordinate System Origin ( Quadrant ) Quadrant Quadrant Quadrant This coordinate system is known as the Cartesian coordinate system. ) is called the origin and denotes the solution The point ( when . The Cartesian coordinate system is made up of above. 4 quadrants labeled Example 2: Plot the following points on the graph above: a.) (3, 2) b.) (-2, 1) c.) (6, -2) d.) (-4, -3) e.) (0, 0) 5 Linear Equations in Two Variables Practice Problems 1. Find at least 2. Does the point (– ) satisfy the equation 3. Does the point (– ) satisfy the equation 4. Plot the following points on the graph below: a.) (– b.) ( solutions to the linear equation ) c.) ( d.) ( ) 6 ? ? ) )