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Linear Equations in Three Variables
R 2 is the space of 2 dimensions. There is an x-coordinate that can be any real number, and there is a y-coordinate that can be any real number. R 3 is the space of 3 dimensions. There is an x, y, and z coordinate. Each coordinate can be any real number.
Linear equations in three variables
If a, b, c and r are real numbers (and if a, b, and c are not all equal to 0) then ax + by + cz = r is called a linear equation in three variables. (The “three variables” are the x, the y, and the z.) The numbers a, b, and c are called the coe cients of the equation. The number r is called the constant of the equation.
Examples. 3 x + 4y 7 z = 2, 2 x + y z = 6, x 17 z = 4, 4y = 0, and x + y + z = 2 are all linear equations in three variables.
Solutions to equations
A solution of a linear equation in three variables ax + by + cz = r is a specific point in R 3 such that when the x-coordinate of the point is multiplied by a, the y-coordinate of the point is multiplied by b, the z-coordinate of the point is multiplied by c, and then those three products are added together, the answer equals r. (There are always infinitely many solutions to a linear equation in three variables.) 255
Linear Equations in Three Variables
R^2 is the space of 2 dimensions. There is an x-coordinate that can be any real number, and there is a y-coordinate that can be any real number. R^3 is the space of 3 dimensions. There is an x, y, and z coordinate. Each coordinate can be any real number.
Linear equations in three variables
If a, b, c and r are real numbers (and if a, b, and c are not all equal to 0) then ax + by + cz = r is called a linear equation in three variables. (The “three variables” are the x, the y, and the z.) The numbers a, b, and c are called the coe cients of the equation. The number r is called the constant of the equation.
Examples. 3 x + 4y 7 z = 2, 2 x + y z = 6, x 17 z = 4, 4y = 0, and x + y + z = 2 are all linear equations in three variables.
Solutions to equations
A solution to a linear equation in three variables ax+by+cz = r is a specific point in R^3 such that when when the x-coordinate of the point is multiplied by a, the y-coordinate of the point is multiplied by b, the z-coordinate of the point is multiplied by c, and then those three products are added together, the answer equals r. (There are always infinitely many solutions to a linear equation in three variables.) 192
Linear Equations in Three Variables
R^2 is the space of 2 dimensions. There is an x-coordinate that can be any real number, and there is a y-coordinate that can be any real number. R^3 is the space of 3 dimensions. There is an x, y, and z coordinate. Each coordinate can be any real number.
Linear equations in three variables
If a, b, c and r are real numbers (and if a, b, and c are not all equal to 0) then ax + by + cz = r is called a linear equation in three variables. (The “three variables” are the x, the y, and the z.)
Examples. 3 x + 4y 7 z = 2, 2 x + y z = 6, x 17 z = 4, 4y = 0, and x + y + z = 2 are all linear equations in three variables.
Solutions to equations
A solution to a linear equation in three variables ax+by+cz = r is a specific point in R^3 such that when when the x-coordinate of the point is multiplied by a, the y-coordinate of the point is multiplied by b, the z-coordinate of the point is multiplied by c, and then those three products are added together, the answer equals r. (There are always infinitely many solutions to a linear equation in three variables.) 1
Linear Equations in Three Variables
JR2 is the space of 2 dimensions. There is an x-coordiuatu IJIHI real number, and there is a y-coordinate that can be any real number. R3 is the space of 3 dimensions. There is an .x, y, and z coordinate. Each coordinate can be any real number.
Linear equations in three variables.
I
If a, b, c and r are real numbers (and if a, b, and c are not all equal to 0)
then ax + by + ez = r is called a linear equation in two variables. (TJir’ “tliv’~
variables” are the x, the y, and the z.)
Examples. 3x+4y—7z=2, —2x+y—z=—6,x—17z=4,4y=0,and x + y + z = 2 are all linear equations in three variables.
Solutions to equations
A solution to a linear equation in three variables ax+by+cz = r is a specific point in JR3 such that when when the x-coordinate of the point is multiplied by a, the y-coordinate of the point is multiplied by b, and the z-coordinate of the point is multiplied by c, and those three numbers are added together, the answer equals r. (There are always infinitely many solutions to a linear equation in three variables.) 1
Linear Equations in Three Variables
JR2 is the space of 2 dimensions. There is an x-coordiuatu IJIHI real number, and there is a y-coordinate that can be any real number. R3 is the space of 3 dimensions. There is an .x, y, and z coordinate. Each coordinate can be any real number.
Linear equations in three variables.
I
If a, b, c and r are real numbers (and if a, b, and c are not all equal to 0)
then ax + by + ez = r is called a linear equation in two variables. (TJir’ “tliv’~
variables” are the x, the y, and the z.)
Examples. 3x+4y—7z=2, —2x+y—z=—6,x—17z=4,4y=0,and x + y + z = 2 are all linear equations in three variables.
Solutions to equations
A solution to a linear equation in three variables ax+by+cz = r is a specific point in JR3 such that when when the x-coordinate of the point is multiplied by a, the y-coordinate of the point is multiplied by b, and the z-coordinate of the point is multiplied by c, and those three numbers are added together, the answer equals r. (There are always infinitely many solutions to a linear equation in three variables.)
Example. The point x = 1, y = 2, and z = 1 is a solution of the equation 2 x + 5y + z = 7 since 2(1) + 5(2) + ( 1) = 2 + 10 1 = 7 The point x = 3, y = 2, and z = 4 is a not a solution of the equation 2 x + 5y + z = 7 since 2(3) + 5( 2) + (4) = 6 10 + 4 = 12 and 12 6 = 7
Linear equations and planes
The set of solutions in R 2 of a linear equation in two variables is a 1- dimensional line. The set of solutions in R 3 of a linear equation in three variables is a 2- dimensional plane.
Solutions of systems of linear equations
As in the previous chapter, we can have a system of linear equations, and we can try to find solutions that are common to each of the equations in the system. We call a solution to a system of equations unique if there are no other solutions. 256
Example. The point x = 1, y = 2, and z = 1 is a solution to the equation 2 x + 5y + z = 7 since 2(1) + 5(2) + ( 1) = 2 + 10 1 = 7 The point x = 3, y = 2, and z = 4 is a not a solution to the equation 2 x + 5y + z = 7 since 2(3) + 5( 2) + (4) = 6 10 + 4 = 12 and 12 ⇥= 7
Linear equations and planes
The set of solutions in R^2 to a linear equation in two variables is a 1- dimensional line. The set of solutions in R^3 to a linear equation in three variables is a 2- dimensional plane.
Solutions to systems of linear equations
As in the previous chapter, we can have a system of linear equations, and we can try to find solutions that are common to each of the equations in the system. We call a solution to a system of equations unique if there are no other solutions. 193
Example. The point x = 1, y = 2, and z = 1 is a solution to the equation 2 x + 5y + z = 7 since 2(1) + 5(2) + ( 1) = 2 + 10 1 = 7 The point x = 3, y = 2, and z = 4 is a not a solution to the equation 2 x + 5y + z = 7 since 2(3) + 5( 2) + (4) = 6 10 + 4 = 12 and 12 6 = 7
Linear equations and planes
The set of solutions in R^2 to a linear equation in two variables is a 1- dimensional line. The set of solutions in R^3 to a linear equation in three variables is a 2- dimensional plane.
Solutions to systems of linear equations
As in the previous chapter, we can have a system of linear equations, and we can try to find solutions that are common to each of the equations in the system. We call a solution to a system of equations unique if there are no other solutions. 2
Example. The point x = 1, y = 2, and z = —1 is a solution to the equation —Zx + 5y + z = 7 since —2(1) + 5(2) + (—1) = —2 + 10 — 1 = 7. The point x = 3, y = —2, and z = 4 is a not a solution to the equation —2x + 5y + z = 7 since —2(3) + 5(—2) + (4) = —6 — 10 + 4 =
—12, and
Linear equations and planes.
The set of solutions in R2 to a linear equation in two variab1r’~ 1 1 - dimensional line.
The set of solutions in F to a linear equation in three variables is a 2-
dimensional plane. A solution to a linear equation in three variables — ax + by + cz = r — is a point in R3 that lies on the plane corresponding to ax + by + cz = r. So we see that there are more solutions to a linear eqnation in three variables (2-dimensions worth) then there are to a linear eqnation in two variables (1-dimensions worth)
ax + C I ‘C
Solutions to systems of linear equations.
As in the previous chapter, we can have a system of linear equations, and ask for a common solution to each equation in the system. If there is a solution to a system of equations, we call that solution unique if there are no other solutions.
It might be that the three planes from the system of three equations will be parallel. Then the three planes wouldn’t intersect. There’d be no point common to all three planes, and hence the system will not have any solutions.
There might not be a point that lies on all three planes even if the planes aren’t parallel. In this case again, there’d be no solution at all.
258
It might be that the three planes from the system of three equations will be parallel. Then the three planes wouldn’t intersect. There’d be no point common to all three planes, and hence the system will not have any solutions.
There might not be a point that lies on all three planes even if the planes aren’t parallel. In this case again, there’d be no solution at all.
It might be that the three planes from the system of three equations will be parallel. Then the three planes wouldn’t intersect. There’d be no point common to all three planes, and hence the system will not have any solutions.
There might not be a point that lies on all three planes even if the planes aren’t parallel. In this case again, there’d be no solution at all.
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VS
It might be that the three planes from the system of three equations will be parallel. Then the three planes wouldn’t intersect. There’d be no point common to all three planes, and hence the system will not have any solutions.
There might not be a point that lies on all three planes even if the planes aren’t parallel. In this case again, there’d be no solution at all.
195
It might be that the three planes from the system of three equations will be parallel. Then the three planes wouldn’t intersect. There’d be no point common to all three planes, and hence the system will not have any solutions.
There might not be a point that lies on all three planes even if the planes aren’t parallel. In this case again, there’d be no solution at all.
4
Sometimes, the three planes will intersect in a way that allows for more than one point to be on all three planes at once. In this case, there are multiple solutions. Because there’s more than one solution, there’s not a unique solution.
Visualize di↵erent arrangements of three planes in R 3 and try to convince yourself that either there is exactly one point contained in all three planes, or no points contained in all three planes, or that there are infinitely many points that are contained in all three planes. That means that a system of three linear equations in three variables will always have either a unique solution, no solution at all, or infinitely many solutions.
259
Sometimes, the three planes will intersect in a way that allows for more than one point to be on all three planes at once. In this case, there are multiple solutions. Because there’s more than one solution, there’s not a unique solution.
Visualize di erent arrangements of three planes in R^3 and try to convince yourself that either there is exactly one point contained in all three planes, or no points contained in all three planes, or that there are infinitely many points that are contained in all three planes. That means that a system of three linear equations in three variables will always have either a unique solution, no solution at all, or infinitely many solutions.
196
Sometimes, the three planes will intersect in a way that allows for more than one point to be on all three planes at once. In this case, there are multiple solutions. Because there’s more than one solution, there’s not a unique solution.
Visualize di erent arrangements of three planes in R^3 and try to convince yourself that either there is exactly one point contained in all three planes, or no points contained in all three planes, or that there are infinitely many points that are contained in all three planes. That means that a system of three linear equations in three variables will always have either a unique solution, no solution at all, or infinitely many solutions.
A
For #11-13, solve the logarithmic equations for x.
11.) log (^) e (2x 3) = 4
12.) log 2 (5x + 1) = 3
13.) log (^) e (x 2 + x) log (^) e (x) = log (^) e (2)
