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:

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.

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Section 3.3 Pg. 1

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 = 07x + 3y + z = 43x − 2y + 6z = 1

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Graph of 3 equations

Graph of 2 equations

Graph of 2 equations

3. Steps for solving a third-order linear system:(a) Write each equation in Ax + By + Cz = D form, without fractions or decimals.

(b) Pick two equations and eliminate one variable. (c) Pica a different set of two equations and eliminate the same variable. (d) Solve the resulting system of two equations in two variables.(e) Back-substitute to find the third variable, and give your answer as an ordered triple.

4. Solve each of the following systems.

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x + 4y − z = 203x + 2y + z = 82x − 3y + 2z = −16

2y − z = 7x + 2y + z = 172x − 3y + 2z = −1

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 = 83x − 2y + 4z = 104x + 2y − 6z = −5

3x + 2y + z = −12x − y − z = 55x + 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?

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Section 3.3 Pg. 3

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:

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( ) = ax2 + bx + c passing through

the points 1,4( ), 2,1( ), and 3,4( ) .

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Section 3.3 Pg. 4

y = f x( ) = ax2 + bx + c

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