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Chapter 5: Problem 4

Determine whether each ordered pair is a solution of the system of equations. \(\left\\{\begin{array}{l}4 x^{2}+y=3 \\ -x-y=11\end{array}\right.\) (a) \((-2,-9)\) (b) \((2,-13)\)

Short Answer

Expert verified
The pair (-2,-9) is not a solution to the system of equations, while the pair (2,-13) is a solution.

Step by step solution

01

Substitute and Test (a)

Put the first pair (-2,-9) into the equations. For the first equation: \(4(-2)^{2}+-9=3\) simplifies to \(16-9= 7\), which doesn鈥檛 equal to 3. So, the pair (-2,-9) doesn鈥檛 satisfy the first equation. We don't need to check the second equation since this ordered pair isn鈥檛 a solution to the system of equations.
02

Substitute and Test (b)

Put the second pair (2,-13) into the equations. For the first equation: \(4(2)^{2}-13=3\) simplifies to \(16-13= 3\), which matches with the right-hand side. Next, for the second equation: \(-2-(-13) = 11\), this simplifies to 11 which matches the right-hand side. So, the pair (2,-13) satisfies both equations, and thus it is a solution to the system of equations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solution Verification
Ensuring that an ordered pair is a solution for a system of equations involves validating if it satisfies all equations within that system. Here's how to do it:
  • Substitution: First, substitute the values of the 'x' and 'y' from the ordered pair directly into each equation of the system.
  • Check Equality: After substitution, perform the arithmetic operations to simplify both sides of each equation.
  • Compare Results: The simplified left-hand side should be equal to the right-hand side of the equation. If it holds true, the ordered pair is a potential solution for that equation.
It's crucial to successfully pass this process for all equations in the system for the ordered pair to be a solution. If even one equation doesn't match, the pair isn't a solution. This steps bring clarity and assurance to whether the solution is correct, providing a solid understanding of the concept.
Algebraic Substitution
Algebraic substitution is a fundamental method in mathematics used to test solutions within equations. It helps in verifying potential solutions by working through a few simple steps:
  • Plugging Values: Insert the 'x' and 'y' values from the ordered pair into the given equation.
  • Calculation: Execute any required arithmetic such as squaring, multiplying, and adding or subtracting.
  • Compare and Simplify: After calculation, compare the results to the constant term or right side of the equation to see if they match.
Substitution is particularly useful for systems of equations because it allows us to check fairly quickly if a given set of values solves the whole system. This technique is not limited to linear equations; it can also apply to quadratic and higher-degree equations.
The ease and straightforwardness of substitution make it an invaluable tool in problem-solving. Being comfortable with substitution helps build a strong mathematical foundation.
Ordered Pairs
Ordered pairs are a way to denote a solution in a two-variable context, typically represented as \((x, y)\). Here's how they function in systems of equations:
  • Relation Representation: The 'x' and 'y' in an ordered pair represent specific values that might solve the equations when substituted.
  • Graphical Interpretation: On a coordinate plane, these pairs indicate points where the two lines intersect, representing solutions to both equations simultaneously.
  • Matching Equations: To verify if an ordered pair is a solution, both 'x' and 'y' must satisfy all equations in the system.
Understood correctly, ordered pairs are not just about numbers in parentheses. They represent possible solutions in a system, which can be analyzed both algebraically and graphically. This concept not only applies to basic algebra, but also expands into more complex systems, aiding in visualization and comprehension.

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Most popular questions from this chapter

Optimal Cost A humanitarian agency can use two models of vehicles for a refugee rescue mission. Each model A vehicle costs $$\$ 1000$$ and each model B vehicle costs $$\$ 1500$$. Mission strategies and objectives indicate the following constraints. \- A total of at least 20 vehicles must be used. 4\. A model A vehicle can hold 45 boxes of supplies. A model B vehicle can hold 30 boxes of supplies. The agency must deliver at least 690 boxes of supplies to the refugee camp. \- A model A vehicle can hold 20 refugees. A model \(\mathrm{B}\) vehicle can hold 32 refugees. The agency must rescue at least 520 refugees. What is the optimal number of vehicles of each model that should be used? What is the optimal cost?

Graphical Reasoning Two concentric circles have radii \(x\) and \(y\), where \(y>x .\) The area between the circles must be at least 10 square units. (a) Find a system of inequalities describing the constraints on the circles. (b) Use a graphing utility to graph the system of inequalities in part (a). Graph the line \(y=x\) in the same viewing window. (c) Identify the graph of the line in relation to the boundary of the inequality. Explain its meaning in the context of the problem.

The given linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the maximum value of the objective function and where it occurs. Objective function: \(z=2.5 x+y\) Constraints: \(\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 3 x+5 y & \leq 15 \\ 5 x+2 y & \leq 10 \end{aligned}\)

Peregrine Falcons The numbers of nesting pairs \(y\) of peregrine falcons in Yellowstone National Park from 2001 to 2005 can be approximated by the linear model \(y=3.4 t+13, \quad 1 \leq t \leq 5\) where \(t\) represents the year, with \(t=1\) corresponding to 2001\. (Sounce: Yellowstone Bird Report 2005) (a) The total number of nesting pairs during this five-year period can be approximated by finding the area of the trapezoid represented by the following system. \(\left\\{\begin{array}{l}y \leq 3.4 t+13 \\ y \geq 0 \\ t \geq 0.5 \\\ t \leq 5.5\end{array}\right.\) Graph this region using a graphing utility. (b) Use the formula for the area of a trapezoid to approximate the total number of nesting pairs.

Sketch the region determined by the constraints. Then find the minimum anc maximum values of the objective function and where they occur, subject to the indicated constraints. Objective function: $$ z=6 x+10 y $$ Constraints: $$ \begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 3 x+5 y & \leq 15 \end{aligned} $$
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