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Linear Equations in Two Variables

Linear Equations in Two VariablesIn this chapter, we ll use the geometry of lines to help us solve Equations in two variablesIfa,b,andrare real numbers (and ifaandbare not both equal to 0) thenax+by=ris called alinear equation in two Variables . (The two Variables are thexand they.)The numbersaandbare called thecoe cientsof the equationax+by= numberris called theconstantof the equationax+by= 3y=5and 2x 4y= 7 are Linear Equations in of equationsAsolutionof a Linear equation in two variablesax+by=ris a specific pointinR2such that when when thex-coordinate of the point is multiplied bya,and they-coordinate of the point is multiplied byb, and those two numbersare added together, the answer equalsr. (There are always infinitely manysolutions to a Linear equation in two Variables .) s look at the equation 2x 3y= thatx=5andy=1isapointinR2that is a solution of thisequation because we can letx=5andy=1intheequation2x 3y=7and then we d have 2(5) 3(1) = 10 3= pointx=8andy=3isalsoasolutionoftheequati on2x 3y=7since 2(8) 3(3) = 16 9= pointx=4andy=6isnotasolutionoftheequatio n2x 3y=7because 2(4) 3(6) = 8 18 = 10, and 106= get a geometric interpretation for what the set of solutions of 2x 3y=7looks like, we can add 3y, subtract 7, and divide by 3 to r

line 5x +2y =4isdierent from slope of the line 2x + y =11. Thosetwo slopes are 5 2 and 2 11 respectively. No solutions. If you reach into a hat and pull out two dierent linear equations in two variables, it’s highly unlikely that the two corresponding lines would have exactly the same slope. But if they did have the same slope, then there 4 5 ...

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