MATH 1029 Exam 1
When solving a system of linear equations by graphing, the system's solution is determined by locating _____.
the intersection point
A solution to a system of linear equations in two variables is an ordered pair that _______.
satisfies both equations in the system
The ordered pair (6,8) is a/an _______ of the inequality x+y>3 because when 6 is substituted for _______ and 8 is substituted for _______ the true statement _______ is obtained.
solution x y 14 > 3
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
The set of all points on one side of a line is called a _____.
half-plane
An algebraic expression in two or more variables describing a quantity that must be maximized or minimized is called a/an _____ function.
objective
In the rectangular coordinate system, the point of intersection of the horizontal axis and the vertical axis is called the ______.
origin
The axes of the rectangular coordinate system divide the plane into regions, called ______. There are ______ of these regions.
quadrants 4
Determine whether the following statement makes sense or does not make sense, and explain your reasoning. An important application of linear programming for businesses involves minimizing profit. A. The statement does not make sense because linear programming is used to help allocate resources to manufacture products in a way that will maximize profit. B. The statement does not make sense because the application of linear programming should instead be used minimize unnecessary loss. C. The statement does not make sense because a business maximizes profit by applying effective marketing strategies. D. The statement does not make sense because a company will not stay in business if it is not making a profit to fund the next venture.
A. The statement does not make sense because linear programming is used to help allocate resources to manufacture products in a way that will maximize profit.
When graphing 3x−6y≥2, to determine which side of the line to shade, choose a test point on 3x−6y=2. A. The statement is false. The test point should be chosen from one of the half-planes. B. The statement is true. The test point should always be on the line. C. The statement is false. The test point should always be chosen to match the inequality. D. The statement is true. It does not matter where the test point is chosen.
A. The statement is false. The test point should be chosen from one of the half-planes.
Determine whether the following statement makes sense or does not make sense, and explain your reasoning. In order to solve a linear programming problem, I use the graph representing the constraints and the graph of the objective function. A. The statement does not make sense because the graph representing the constraints is not required. B. The statement does not make sense because the graph of the objective function is not required. C. The statement makes sense because the optimal solution is represented by the intersection of the two graphs. D. The statement makes sense because the graph of the objective function is used to find the corner points and the graph representing the constraints is used to determine which one is the best.
B. The statement does not make sense because the graph of the objective function is not required.
The graph of the linear equation 3x−2y=4 is used to graph the linear inequality 3x−2y>4. A. The statement is false. The linear equation 3x−2y=4 is separate from the solution to the linear inequality 3x−2y>4. B. The statement is true. The linear equation 3x−2y=4 divides the points that satisfy the linear inequality 3x−2y>4 from the rest of the points. C. The statement is false. The linear equation 3x−2y=4 determines the points that satisfy the linear inequality 3x−2y>4. D. The statement is true. The linear equation 3x−2y=4 is included in the solution to the linear inequality 3x−2y>4.
B. The statement is true. The linear equation 3x−2y=4 divides the points that satisfy the linear inequality 3x−2y>4 from the rest of the points.
Explain how to solve a system of equations using graphing. A. To solve a system of linear equations by graphing, graph both equations in the same rectangular coordinate system. For a system with one solution, the y-coordinates of the y-intercept of each line is the system's solution. B. To solve a system of linear equations by graphing, graph both equations in the same rectangular coordinate system. For a system with one solution, the coordinates of the point of intersection of the lines is the system's solution. C. To solve a system of linear equations by graphing, solve either of the equations for one variable in terms of the other. Substitute the expression into the other equation. Solve the resulting equation containing one variable. Back-substitute the value found into one of the original equations.
B. To solve a system of linear equations by graphing, graph both equations in the same rectangular coordinate system. For a system with one solution, the coordinates of the point of intersection of the lines is the system's solution.
What is the solution to a system of linear equations? A. An ordered pair that satisfies at least one equation in the system B. A value of y C. A value of x D. An ordered pair that satisfies both equations in the system
D. An ordered pair that satisfies both equations in the system
A method for finding the maximum or minimum value of a quantity that is subject to various limitations is called _____.
linear programming
The rectangular coordinate system is formed using two number lines that intersect at ______ angles at their zero points. The _______ line is the x-axis and the _______ line is the y-axis. Their point of intersection, _____, is the origin. Each point in the system corresponds to an ordered pair of real numbers, (x,y).
right horizontal vertical (0,0)
The ordered pair (5, 31) is a/an ______ of the equation 11y=4x+11 because when 5 is substituted for x and 31 is substituted for y, a true statement is obtained. Also, it is said that (5, 31) ______ the equation.
solution satisfies
What is the disadvantage to solving a system of equations using the graphing method? A. A solution with complicated fractions would be difficult to "see" as an intersection point on a graph. B. The solution is sometimes not at the intersection point. C. One of the other solution methods must be used to obtain equations that can be used for the graphing method, meaning that extra work needs to be done. D. If there is no intersection point, the actual solution is unclear.
A. A solution with complicated fractions would be difficult to "see" as an intersection point on a graph.
Determine whether the following statement makes sense or does not make sense, and explain your reasoning. I use the coordinates of each vertex from my graph representing the constraints to find the values that maximize or minimize an objective function. A. The statement makes sense because the coordinates of the vertices represent the values of the variables in the objective function. B. The statement does not make sense because the maximum or minimum value of the objective function might not be at the vertices. C. The statement does not make sense because the graph of the objective function is used to find the vertices. D. The statement makes sense because the coordinates of the vertices represent the value of the objective function.
A. The statement makes sense because the coordinates of the vertices represent the values of the variables in the objective function.
The graph of 5x−2y>3 includes the line 5x−2y=3. A. The statement is true. The graph of an inequality with the > symbol includes the line, while the ≥ symbol does not include the line. B. The statement is true. The graph of an inequality always includes the line. C. The statement is false. The graph of an inequality never includes the line. D. The statement is false. The graph of an inequality with the > symbol does not include the line, while the ≥ symbol includes the line.
D. The statement is false. The graph of an inequality with the > symbol does not include the line, while the ≥ symbol includes the line.
The set of all points that satisfy an inequality is called the _______ of the inequality.
graph
To find the y-intercept of an equation, set ___ = ___ and solve the equation for ___.
x = 0 y
In the rectangular coordinate system, the horizontal number line is called the _______.
x-axis
The first number in an ordered pair is called the _______ and the second term is called the _______.
x-coordinate y-coordinate
The x-coordinate of the point where a graph crosses the x-axis is called the _______.
x-intercept
To find the x-intercept of an equation, set ___ = ___ and solve the equation for ___.
y = 0 x
In the rectangular coordinate system, the vertical number line is called the _______.
y-axis
The point (0,3) lies along a line, so 3 is a/an _________ of that line.
y-intercept
