 Solving Algebraically COMPARISON equal to the same thing, they must be equal to each other. This principle will allow us to mix the equations a different way. Example 1: y = 2x + 13 y = 4x - 3 Since y is equal to both 2x + 13 and 4x - 3, those things must be equal: 2x + 13 = 4x - 3 Now solve: 2x - 4x = -3 - 13 -2x = -16 x = 8 Now substitute this in either equation to find the matching y value: Using the first equation: y = 2(8) + 13 = 16 + 13 = 29 Solution is (8, 29) Example 2: 10x - 8y = 12 10x = 4y + 15 Here, with a slight rearrangement of the first equation, we get: 10x = 8y + 12 10x = 4y + 15 Since 10x is equal to both 8y + 12 and 4y + 15, these must equal each other: 8y + 12 = 4y + 15 Now solve: 8y - 4y = 15 - 12 4y = 3 y = 3/4 = 0.75 Substitute this in the first original equation: 10x - 8(0.75) = 12 10x - 6 = 12 10x = 18 x = 1.8 Solution is (1.8, 0.75) The advantages of solving systems algebraically over graphing are illustrated on this page. Graphing is already much more work. In addition, in the first example, the intersection point (8, 29) likely would have been off the graph you plotted, necessitating replotting it on a larger scale. Also, in the second example, the exact solution (1.8, 0.75) would not likely have been your estimate if determined from a graph, as ordinary graph paper doesn't use such precision; your guess would have been off a little. Having a rounded, 'slightly off' answer also makes checking problematic, as a rounded answer won't work; you won't know if this is because the answer is wrong, or just close. For these reasons, solving by graphing is not the best choice, and in practice is seldom used. |