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DeSmet - Math 152 Blitzer 5E ∫ 3.3 - Systems of Linear Equations in Three Variables 1. Linear Equation in Three Variables: An example of a linear equation in three variables is 2x + 3y − z = 10 . A solution to this linear equation would be an ordered triple of the form ( x, y, z ) . Example solutions ( 2, 2, 0 ) , ( 0, 0, −10 ) , and ( 7, 0, 4 ) . Do you see why they are solutions? Such solutions live in “three-space” and when the solutions are graphed, the graph is a plane. (A quick note, you can think of solving for z and you would get z as a function of x and y, i.e. z = f ( x, y ) = 2x + 3y − 10 . This may help with the plane idea...) 2. Systems of Linear Equations in Three Variables: Consider the system of linear equations in three variables: Each equation is a plane in space, so to solve this “third-order system,” we are finding the point or points that are in all three planes at the same time. x+ y+z = 0 7x + 3y + z = 4 3x − 2y + 6z = 1 Possible scenarios: Tell whether the solution set of the graphed system of three linear equations in three variables has one ordered triple, many ordered triples, or no solutions. z z z y y y Graph of 3 equations x x x Graph of 2 equations z z z Graph of 2 equations y y x x x z z x y y Section 3.3 x x Pg. 1 DeSmet - Math 152 Blitzer 5E 3. Steps for solving a third-order linear system: (a) Write each equation in Ax + By + Cz = D form, without fractions or decimals. (b) (c) (d) (e) Pick two equations and eliminate one variable. Pica a different set of two equations and eliminate the same variable. Solve the resulting system of two equations in two variables. Back-substitute to find the third variable, and give your answer as an ordered triple. 4. Solve each of the following systems. x + 4y − z = 20 3x + 2y + z = 8 2x − 3y + 2z = −16 Section 3.3 2y − z = 7 x + 2y + z = 17 2x − 3y + 2z = −1 Pg. 2 DeSmet - Math 152 Blitzer 5E 5. Inconsistent and Dependent Systems: When solving such systems, you can encounter a situation when all the variables disappear. If this happens you have either no solutions ( 2 = 8 ) , or infinitely many solutions ( 2 = 2 ) . Infinitely many solutions may be a plane, or a line in 3-space. Solve each system below: 2x + y − 3x = 8 3x − 2y + 4z = 10 4x + 2y − 6z = −5 3x + 2y + z = −1 2x − y − z = 5 5x + y = 4 6. Applications: Recent studies indicate that a child’s intake of cholesterol should be no more than 300 mg per day. By eating 1 egg, 1 cupcake, and 1 slice of pizza, a child consumes 302 mg of cholesterol. A child who eats 2 cupcakes and 3 slices of pizza takes in 65 mg of cholesterol. By eating 2 eggs and 1 cupcake, a child consumes 567 mg of cholesterol. How much cholesterol is in each item? Section 3.3 Pg. 3 DeSmet - Math 152 Blitzer 5E 7. Curve Fitting: Pick three points in the plane. Most likely these points do not lie on a line. I can fit a curve to these points though, I can fit a function of the form: y = f ( x ) = ax 2 + bx + c This is called a quadratic function, which we will study in depth in chapter 8. A typical graph of a quadratic function looks like a bowl (see the example below). Find the equation of the quadratic function of the from y = f ( x ) = ax 2 + bx + c passing through the points (1, 4 ) , ( 2,1) , and ( 3, 4 ) . 6 5 4 3 2 1 0 Section 3.3 1 2 3 4 Pg. 4