5. Systems of Equations and Inequalities Flashcards
A solution to a system of linear equations in two variables is an ordered pair that _____.
satisfies both equations in the system
When solving a system of linear equations by graphing, the system’s solution is determined by locating _____.
the intersection point
When solving
{ 3x – 2y = 5
{ y = 3x – 3
by the substitution method, we obtain x = 1/3, so the solution set is _____.
{(1/3, –2)}
When solving
{ 2x + 10y = 9
{ 8x + 5y = 7
by the addition method, we can eliminate y by multiplying the second equation by _____ and then adding the equations.
–2
When solving
{ 4x – 3y = 15
{ 3x – 2y = 10
by the addition method, we can eliminate y by multiplying the first equation by 2 and the second equation by _____ and then adding the equations.
–3
When solving
{ 12x – 21y = 24
{ 4x – 7y = 7
by the addition method, we obtain 0 = 3, so the solution set is _____.
The linear system is a/an _____.
If you attempt to solve such a system by graphing, you will obtain two lines that are _____.
∅
inconsistent
parallel
When solving
{ x = 3y + 2
{ 5x – 15y = 10
by the substitution method, we obtain 10 = 10, so the solution set is _____.
The equations in this system are called ______.
If you attempt to solve such a system by graphing, you will obtain two lines that ______.
{(x, y) | x = 3y + 2}
or
{(x, y) | 5x – 15 = 10}
dependent
are identical or coincide
A company’s _____ function is the money generated by selling x units of its product.
The difference between this function and the company’s cost function is called its _____ function.
revenue
profit
A company has a graph that shows the money it generates by selling x units of its product.
It also has a graph that shows its cost of producing x units of its product.
The point of intersection of these graphs is called the company’s _____.
break-even point
A solution of a system of linear equations in three variables is an ordered _____ of real numbers that satisfies all?/some? (choose the correct choice) of the equations in the system.
triple
all
Consider the following system:
{ x + y – z = –1
{ 2x – 2y – 5z = 7
{ 4x + y – 2z = 7
We can eliminate x from Equations 1 and 2 by multiplying Equation 1 by _____ and adding equations.
We can eliminate x from Equations 1 and 3 by multiplying Equation 1 by _____ and adding equations.
–2
–4
Consider the following system:
{ x + y + z = 2
{ 2x – 3y = 3
{ 10y – z = 12
Equation 2 does not contain the variable _____.
To obtain a second equation that does not contain this variable, we can ______.
z
add Equations 1 and 3
Correct or incorrect:
[7x / (x + 2)(x – 3)] = [A / (x + 2)] + [B / (x – 3)]
Correct
Correct or incorrect
[3x / (x+5)(x – 4)²] = [A / (x + 5)] + [B / (x – 3)
Incorrect
Correct or incorrect
[1 / (x + 1)(x² + 4)] = [A / (x + 1)] + [B / (x² + 4)]
Incorrect
Correct or incorrect
[(7x – 5) / (x² + x + 1)²] = [(Ax + B) / (x² + x + 1)] + [(Cx + D) / (x² + x + 1)²]
Correct
A system of two equations in two variables that contains at least one equation that cannot be expressed in the form Ax + By = C is called a system of _____ equations.
nonlinear
When solving
{ x² – 4y = 4
{ x + y = –1
by the substitution method, we obtain x = –4 or x = 0, so the solution set is _____.
{(–4, 3), (0, 1)}
When solving
{ 3x² + 2y² = 35
{ 4x² + 3y² = 48
by the addition method, we can eliminate x² by multiplying the first equation by –4 and the second equation by _____ and then adding the equations.
3
When solving
{ x² + 4y² = 16
{ x² – y² = 1
by the addition method, we obtain y² = 3, so the solution set is _____.
{(2, √3), (2, –√3), (–2, √3), (–2, –√3)},
When solving
{ x² + y² = 13
{ x² – y = 7
by the addition method, we can eliminate x² by multiplying the second equation by _____ and then adding the equations.
We obtain ______, a quadratic equation in y.
–1
y² + y = 6
When solving
{ x² + 4y² = 20
{ xy = 4
by the substitution method, we can eliminate y by solving the second equation for y. We obtain y = _____.
Then we substitute _____ for _____ in the first equation.
4 / x
4 / x
y
The ordered pair (3, 2) is a/an _____ of the inequality x + y > 1 because when 3 is substituted for _____ and 2 is substituted for _____, the true statement ______ is obtained.
solution
x
y
5 > 1
The set of all points that satisfy a linear inequality in two variables is called the _____ of the inequality.
graph
The set of all points on one side of a line is called an/an ______.
half-plane
True or false
The graph of 2x – 3y > 6 includes the line 2x – 3y = 6.
false
True or false
The graph of the linear equation 2x – 3y = 6 is used to graph the linear inequality 2x – 3y > 6.
true
True or false
When graphing 4x – 2y ≥ 8, to determine which side of the line to shade, choose a test point on 4x – 2y = 8
false
When graphing x² + y² > 25, to determine whether to shade the region inside the circle or the region outside the circle, we can use _____ as a test point.
(0, 0), although answers will vary
The solution set of the system
{ x – y < 1
{ 2x + 3y ≥ 12
is the set of ordered pairs that satisfies _____ and ______.
x – y < 1
2x + 3y ≥ 12
True or false
The graph of the solution set of the system
{ x – 3y < 6
{ 2x + 3y ≥ –6
includes the intersection point of x – 3y = 6 and 2x + 3y = –6.
false
A method for finding the maximum or minimum value of a quantity that is subject to various limitations is called ______.
linear programming
An algebraic expression in two or more variables describing a quantity that must be maximized or minimized is called a/an ______ function.
objective
A system of linear inequalities is used to represent restrictions, or _____, on a function that must be maximized or minimized.
Using the graph of such a system of inequalities, the maximum and minimum values of the function occur at one or more of the _____ points.
constraints
corner
