Problem 37
Question
Solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(\left\\{\begin{array}{r}{x+3 y=2} \\ {3 x+9 y=6}\end{array}\right.\)
Step-by-Step Solution
Verified Answer
The given linear system has infinitely many solutions. The solution set in set notation is \(\{(x,y) | x = 2 - 3y\}\). This means all pairs (x,y) where x equals 2 - 3y are solutions to the system.
1Step 1: Identify and Confirm
First, note that the second equation is indeed 3 times the first equation: \n\[3x + 3*3y = 2*3\] which simplifies to \n\[3x + 9y = 6\] So, yes, the second equation is a multiple of the first equation, thus every solution to the first equation is also a solution to the second equation.
2Step 2: Solve One of the Equations
Solve the first equation for 'x' to find its relationship with 'y'. Rearrange \[x + 3y = 2\] to be \[x = 2 - 3y\] This means the value of 'x' is determined entirely by the value of 'y'.
3Step 3: Solution Set Using Set Notation
Express the solution set in set notation: \[\{(x,y) | x = 2 - 3y\}\] Remember, this denotes the set of all possible (x,y) pairs where 'x' equals '2 - 3y' which is compatible with the equation.
Key Concepts
systems with infinitely many solutionsset notationlinear equationsidentifying equivalent equations
systems with infinitely many solutions
When we deal with systems of linear equations, sometimes we will encounter systems that have infinitely many solutions. This happens when both equations in the system represent the same line. It means that every point on that line is a solution to the system.
For instance, in our exercise, the given system of equations
The second equation is simply a multiple of the first, indicating they are effectively the same line. Hence, any solution for one equation will satisfy the other, resulting in an infinite number of solutions that lie along this line.
For instance, in our exercise, the given system of equations
- \(x + 3y = 2\)
- \(3x + 9y = 6\)
The second equation is simply a multiple of the first, indicating they are effectively the same line. Hence, any solution for one equation will satisfy the other, resulting in an infinite number of solutions that lie along this line.
set notation
Set notation is a mathematical language used to describe sets, or groups, of numbers. This notation is useful when expressing the solutions of equations precisely.
In the context of our problem, the solution set in set notation is represented as:
Set notation makes it clear that instead of a single solution, there are many pairs \((x, y)\) that could satisfy the system because the solution is dependent on the chosen value of \(y\). As you choose different values for \(y\), you will get corresponding \(x\) values, forming the whole set of solutions.
In the context of our problem, the solution set in set notation is represented as:
- \(\{(x,y) \mid x = 2 - 3y\}\)
Set notation makes it clear that instead of a single solution, there are many pairs \((x, y)\) that could satisfy the system because the solution is dependent on the chosen value of \(y\). As you choose different values for \(y\), you will get corresponding \(x\) values, forming the whole set of solutions.
linear equations
Linear equations are equations between two variables that produce a straight line when plotted on a graph. Each linear equation can be written in the form \(ax + by = c\), where \(x\) and \(y\) are variables, and \(a\), \(b\), and \(c\) are constants.
In the exercise:
These equations are fundamental because they can represent various real-world situations using straightforward expressions. Understanding their graphical representation will help you see why some systems don't have a unique solution but instead infinitely many solutions or sometimes no solutions at all.
In the exercise:
- \(x + 3y = 2\)
- \(3x + 9y = 6\)
These equations are fundamental because they can represent various real-world situations using straightforward expressions. Understanding their graphical representation will help you see why some systems don't have a unique solution but instead infinitely many solutions or sometimes no solutions at all.
identifying equivalent equations
Equivalent equations are equations that have the same solutions. Identifying these in a system is key to understanding the nature of the solutions. In our exercise, the equation \(3x + 9y = 6\) is equivalent to the equation \(x + 3y = 2\) because it is simply the first equation multiplied by three.
To determine whether two equations are equivalent:
This concept helps in recognizing when a system has infinitely many solutions or is composed of different lines leading to no solutions, based on whether the lines represented by the equations intersect or overlap perfectly.
To determine whether two equations are equivalent:
- Simplify both equations as much as possible.
- See if one equation can be obtained from the other through multiplication by a constant.
This concept helps in recognizing when a system has infinitely many solutions or is composed of different lines leading to no solutions, based on whether the lines represented by the equations intersect or overlap perfectly.
Other exercises in this chapter
Problem 36
Use the four-step strategy to solve each problem. Use \(x, y,\) and \(z\) to represent unknown quantities. Then translate from the verbal conditions of the prob
View solution Problem 36
write the partial fraction decomposition of each rational expression. $$\frac{3 x^{2}-2 x+8}{x^{3}+2 x^{2}+4 x+8}$$
View solution Problem 37
Solve each system by the method of your choice. $$ \left\\{\begin{array}{l} {x^{2}+(y-2)^{2}=4} \\ {x^{2}-2 y=0} \end{array}\right. $$
View solution Problem 37
Use the four-step strategy to solve each problem. Use \(x, y,\) and \(z\) to represent unknown quantities. Then translate from the verbal conditions of the prob
View solution