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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
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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
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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
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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
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Pg. 4