Download 1 Lines 2 Linear systems of equations

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Lines
Line equations
Given
Point (x1 , y1 ), slope m
Two points (x1 , y1 ), (x2 , y2 )
m-undefined
Slope m, y-intercept (0, b)
Use
Equation
Point-slope form
y − y1 = m(x − x1 )
If x1 = x2 , the line is vertical
x = x1
If x1 ̸= x2 , then m is:
−y1
m = xy22 −x
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Then use point-slope form
y − y1 = m(x − x1 )
Line is horizontal
y = y1
Slope-intercept form
y = mx + b
Pairs of lines
Pair of lines
Both vertical
The lines are
a) Coincident, if they have the same x-intercept
b) Parallel, if they have different x-intercept
One vertical, one non-vertical a) Intersecting
b) Parallel, if the non-vertical line is horizontal
Neither is vertical
Write the equations in slope-intercept form
y = m1 x + b1 and y = m2 x + b2
a) Coincident, if m1 = m2 and b1 = b2
b) Parallel, if m1 = m2 , but b1 ̸= b2
c) Intersecting, if m1 ̸= m2
d) Perpendicular, if m1 m2 = −1.
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Linear systems of equations
Steps for solving by substitution
STEP 1 Pick one of the equations and solve for one of the variables in terms of the remaining
variables.
STEP 2 Substitute the result in the remaining equations.
STEP 3 If one equation in one variable results, solve this equation. Otherwise, repeat Steps 1 and
2 until a single equation with one variable remains.
STEP 4 Find the values of the remaining variables by back-substitution.
STEP 5 Check the solution found.
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Rules for Obtaining an Equivalent System of Equations
1. Interchange any two equations in the system.
2. Multiply (or divide) each side of an equation by the same nonzero constant.
3. Replace any equation in the system by the sum (or difference) of that equation and a
nonzero multiple of any other equation in the system.
Steps for solving by elimination
STEP 1 Select two equations from the system and eliminate a variable from them.
STEP 2 If there are additional equations in the system, pair off equations and eliminate the same
variable from them.
STEP 3 Continue Steps 1 and 2 on successive systems until one equation containing one variable
remains.
STEP 4 Solve for this variable and back-substitute in previous equations until all the variables have
been found.
Matrices
Row Operations
1. Interchange any two rows.
2. Replace a row by a nonzero multiple of that row.
3. Replace a row by the sum of that row and a constant nonzero multiple of some other row.
A matrix is in row-echelon form when
1. The entry in row 1, column 1 is a 1, and 0s appear below it.
2. The first nonzero entry in each row after the first row is a 1, 0s appear below it, and it
appears to the right of the first nonzero entry in any row above.
3. Any rows that contain all 0s to the left of the vertical bar appear at the bottom.
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Matrix Method for Solving a System of Linear Equations (Row Echelon Form)
STEP 1 Write the augmented matrix that represents the system.
STEP 2 Perform row operations that place the entry 1 in row 1, column 1.
STEP 3 Perform row operations that leave the entry 1 in row 1, column 1 unchanged, while causing
0s to appear below it in column 1.
STEP 4 Perform row operations that place the entry 1 in row 2, column 2, but leave the entries in
columns to the left unchanged. If it is impossible to place a 1 in row 2, column 2, then
proceed to place a 1 in row 2, column 3. Once a 1 is in place, perform row operations to
place 0s below it.
(If any rows are obtained that contain only 0s on the left side of the vertical bar, place
such rows at the bottom of the matrix.)
STEP 5 Now repeat Step 4, placing a 1 in the next row, but one column to the right. Continue
until the bottom row or the vertical bar is reached.
STEP 6 The matrix that results is the row-echelon form of the augmented matrix. Analyze the
system of equations corresponding to it to solve the original system.
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Systems of Linear Inequalities
Steps for Graphing a Linear Inequality
STEP 1 Graph the corresponding linear equation, a line L. If the inequality is non-strict, graph L
using a solid line; if the inequality is strict, graph L using dashes.
STEP 2 Select a test point P not on the line L.
STEP 3 Substitute the coordinates of the test point P into the given inequality. If the coordinates
of this point P satisfy the linear inequality, then all points on the same side of L as the
point P satisfy the inequality. If the coordinates of the point P do not satisfy the linear
inequality, then all points on the opposite side of L from P satisfy the inequality.
Linear Programming Problem A linear programming problem in two variables, x and y,
consists of maximiz- ing or minimizing an objective function
z = Ax + By
where A and B are given real numbers, not both zero, subject to certain conditions or constraints expressible as a system of linear inequalities in x and y.
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Existence of a Solution Consider a linear programming problem with the set R of feasible
points and objective function z = Ax + By.
1. If R is bounded, then z has both a maximum and a minimum value on R.
2. If R is unbounded and A ≥ 0, B ≥ 0, and the constraints include x ≥ 0 and y ≥ 0, then z
has a minimum value on R but not a maximum value (see Example 2).
3. If R is the empty set, then the linear programming problem has no solution and z has
neither a maximum nor a minimum value.
Fundamental Theorem of Linear Programming If a linear programming problem has a
solution, it is located at a corner point of the set of feasible points; if a linear programming
problem has multiple solutions, at least one of them is located at a corner point of the set of
feasible points. In either case the corresponding value of the objective function is unique.
Steps for Solving a Linear Programming Problem If a linear programming problem has
a solution, follow these steps to find it:
STEP 1 Write an expression for the quantity that is to be maximized or minimized (the objective
function).
STEP 2 Determine all the constraints and graph the set of feasible points.
STEP 3 List the corner points of the set of feasible points.
STEP 4 Determine the value of the objective function at each corner point.
STEP 5 Select the maximum or minimum value of the objective function.
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