881 9.1 Systems of Linear Equations Points of a line in common II I Variation of Figure 6(c) Application of Systems to Model Data Applications with three unknowns usually require solving a system of three equations. If we know three points on the graph, we can find the equation of a parabola in the form y =ax2 +bx +c by solving a system of three equations with three variables. EXAMPLE 7 Solving a System of Two Equations with Three Variables Solve the system. x + 2y + z = 4 (1) 3x - y - 4z = -9 (2) SOLUTION Geometrically, the solution is the intersection of the two planes given by equations (1) and (2). The intersection of two different nonparallel planes is a line, which can be represented as a variation of Figure 6(c). Thus there will be an infinite number of ordered triples in the solution set, representing the points on the line of intersection. To eliminate x, multiply both sides of equation (1) by -3 and add the result to equation (2). (Either y or z could have been eliminated instead.) -3x - 6y - 3z = -12 Multiply (1) by -3. 3x - y - 4z = -9 (2) -7y - 7z = -21 (3) -7y = 7z - 21 Add 7z. y = -z + 3 Divide each term by -7. This gives y in terms of z. Express x also in terms of z by solving equation (1) for x and substituting -z + 3 for y in the result. x + 2y + z = 4 (1) x = -2y - z + 4 Solve for x. x = -21-z + 32 - z + 4 Substitute -z + 3 for y. x = 2z - 6 - z + 4 Distributive property x = z - 2 Combine like terms. The system has an infinite number of solutions. For any value of z, the value of x is z - 2 and the value of y is -z + 3. For example, if z = 2, then x = 2 - 2 = 0 and y = -2 + 3 = 1, giving the solution 10, 1, 22. Verify that another solution is 1-1, 2, 12. With z arbitrary, the solution set is of the form 51z - 2, -z + 3, z26. S Now Try Exercise 59. Solve this equation for y. Use parentheses around -z + 3. NOTE Had we solved equation (3) in Example 7 for z instead of y, the solution would have had a different form but would have led to the same set of solutions.51-y + 1, y, -y + 326 Solution set with y arbitrary If we chose y = 1, one solution would be 10, 1, 22, which was found above.
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