Problem 16
Question
Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}x+2 y=2 \\ x-y=2\end{array}\right.$$
Step-by-Step Solution
Verified Answer
The solution to the system of equations is \(\{(2,0)\}\).
1Step 1: Express Each Equation in y = mx + b Form
Firstly, we must rearrange each equation into the y = mx + b format, where m represents the slope and b is the y-intercept. For the equation \(x + 2y = 2\), subtract x from both sides to get \(2y = -x + 2\), then divide each side by 2 to get \(y = -0.5x + 1\). For the equation \(x - y = 2\), subtract x from both sides to get \(-y = -x + 2\), and then multiply each side by -1 to get \(y = x - 2\).
2Step 2: Graph Each Equation
Next, graph each line on the same set of axes. The line of the first equation \(y = -0.5x + 1\) has a slope of -0.5 and y-intercept of 1. So it starts from point (0,1) and goes down by 1 for every 2 steps to the right. The line of the second equation \(y = x - 2\) has a slope of 1 and y-intercept of -2, starting from point (0,-2) it goes up by 1 for every step to the right.
3Step 3: Identify Point of Intersection
The solution to the system of equations is the point where the two lines intersect. In this case, the two lines intersect at (2,0).
4Step 4: Write the Solution in Set Notation
The solution set in set notation is \(\{(2,0)\}\). This means that x=2 and y=0 are the solutions to the system of equations.
Key Concepts
System of Linear EquationsGraphing Linear EquationsSlope-Intercept FormSet Notation
System of Linear Equations
When we talk about a system of linear equations, we're referring to a collection of two or more linear equations that we solve together. The variables in each equation typically overlap, and our goal is to find the values of these variables that will satisfy all equations in the system simultaneously.
The concept is like a puzzle where every equation is a clue. To solve it, we look for a point (or points) where the lines representing each equation intersect on a graph, indicating that the x and y values at that point hold true for all equations involved. When we find this point, we say that we have found the solution to the system. If the lines don't intersect at any point, this system has no solution, while if the lines coincide, it means there is an infinite number of solutions because the lines are essentially the same.
The concept is like a puzzle where every equation is a clue. To solve it, we look for a point (or points) where the lines representing each equation intersect on a graph, indicating that the x and y values at that point hold true for all equations involved. When we find this point, we say that we have found the solution to the system. If the lines don't intersect at any point, this system has no solution, while if the lines coincide, it means there is an infinite number of solutions because the lines are essentially the same.
Graphing Linear Equations
The task of graphing linear equations is essential for visualizing the relationship between two variables. Each equation can be graphed as a straight line on a coordinate plane.
How do we graph a linear equation? Firstly, we need two points to draw a line. Often, we begin with the y-intercept, where the line crosses the y-axis, and then use the slope to find another point. The slope tells us how steep the line is - it's the ratio of the vertical 'rise' to the horizontal 'run' between two points on the line. A positive slope means the line inclines upwards, while a negative slope means it declines. By plotting the y-intercept and using the slope to find another point, we can graph the entire line by extending it in both directions.
How do we graph a linear equation? Firstly, we need two points to draw a line. Often, we begin with the y-intercept, where the line crosses the y-axis, and then use the slope to find another point. The slope tells us how steep the line is - it's the ratio of the vertical 'rise' to the horizontal 'run' between two points on the line. A positive slope means the line inclines upwards, while a negative slope means it declines. By plotting the y-intercept and using the slope to find another point, we can graph the entire line by extending it in both directions.
Slope-Intercept Form
One of the friendliest formats for graphing is the slope-intercept form, which is written as y = mx + b. Here, 'm' represents the slope of the line, and 'b' is the y-intercept, the point where the line crosses the y-axis.
By placing a linear equation in this form, you instantly get the slope and y-intercept, making it super simple to graph. This form is incredibly intuitive and powerful – the slope tells us how to move from the y-intercept to another point on the line. A positive 'm' means the line rises as it moves to the right, while a negative 'm' means it falls. The y-intercept 'b' puts us on the graph right away without any calculations, which is why many prefer this format when tackling linear equations.
By placing a linear equation in this form, you instantly get the slope and y-intercept, making it super simple to graph. This form is incredibly intuitive and powerful – the slope tells us how to move from the y-intercept to another point on the line. A positive 'm' means the line rises as it moves to the right, while a negative 'm' means it falls. The y-intercept 'b' puts us on the graph right away without any calculations, which is why many prefer this format when tackling linear equations.
Set Notation
In mathematics, we often use set notation to describe collections of numbers or solutions. It's like a mathematical version of a list. When we solve a system of equations, we're actually finding the set of all points (x, y) that make all equations true.
For example, if we have a single solution, we write it as \(\{(x,y)\}\), with the x and y values inside curly braces. This tells us these numbers are members of the set of solutions. If there is no solution, the set is empty, represented as \(\emptyset\). For an infinite number of solutions, which happens when the lines are the same, we represent the solution set with the equation of the line itself, since any point on that line is a valid solution for the system of equations.
For example, if we have a single solution, we write it as \(\{(x,y)\}\), with the x and y values inside curly braces. This tells us these numbers are members of the set of solutions. If there is no solution, the set is empty, represented as \(\emptyset\). For an infinite number of solutions, which happens when the lines are the same, we represent the solution set with the equation of the line itself, since any point on that line is a valid solution for the system of equations.
Other exercises in this chapter
Problem 16
In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to expre
View solution Problem 16
A rectangular lot whose perimeter is 1600 feet is fenced along three sides. An expensive fencing along the lot's length costs 20 dollar per foot. An inexpensive
View solution Problem 16
Graph the solution set of each system of linear inequalities. $$\left\\{\begin{array}{l}y \geq 3 x-2 \\\y \leq 3 x+1\end{array}\right.$$
View solution Problem 16
Solve each system by the substitution method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $
View solution