Solve The System $y = 2x + 6$ And $3y = 6x + 18$ Using Graph Paper Or Graphing Technology.What Is The Solution To The System?A. (2, 6) B. (3, 6) C. No Solutions D. Infinite Solutions
Introduction
In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system. In this article, we will solve a system of two linear equations using graph paper or graphing technology.
The System of Linear Equations
The system of linear equations we will be solving is:
Graphing the Equations
To solve the system of linear equations, we can graph the two equations on a coordinate plane. The first equation is a linear equation in slope-intercept form, which is , where is the slope and is the y-intercept. The second equation is also a linear equation, but it is not in slope-intercept form.
Graphing the First Equation
The first equation is . To graph this equation, we can use the slope-intercept form of a linear equation, which is . In this case, the slope is 2 and the y-intercept is 6.
# Graphing the First Equation
## Slope-Intercept Form
The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
## Graphing the Equation
To graph the equation y = 2x + 6, we can start by plotting the y-intercept, which is (0, 6). Then, we can use the slope to find another point on the line. Since the slope is 2, we can move 1 unit to the right and 2 units up from the y-intercept to find another point on the line.
## Plotting the Points
The two points on the line are (0, 6) and (1, 8). We can plot these points on a coordinate plane and draw a line through them to represent the equation y = 2x + 6.
Graphing the Second Equation
The second equation is . To graph this equation, we can first divide both sides of the equation by 3 to get . This is the same equation as the first equation, so we can use the same graph.
# Graphing the Second Equation
## Dividing Both Sides by 3
To graph the equation 3y = 6x + 18, we can first divide both sides of the equation by 3 to get y = 2x + 6. This is the same equation as the first equation, so we can use the same graph.
## Using the Same Graph
Since the two equations are the same, we can use the same graph to represent both equations. The graph of the second equation is the same as the graph of the first equation.
Finding the Solution
To find the solution to the system of linear equations, we need to find the point of intersection between the two graphs. The point of intersection is the point where the two lines cross.
# Finding the Solution
## Point of Intersection
The point of intersection between the two graphs is the point where the two lines cross. To find this point, we can set the two equations equal to each other and solve for x.
## Setting the Equations Equal
Since the two equations are the same, we can set them equal to each other and solve for x. This gives us the equation 2x + 6 = 2x + 6.
## Solving for x
Solving for x, we get x = 3.
## Finding the Value of y
Now that we have the value of x, we can substitute it into one of the original equations to find the value of y. Substituting x = 3 into the first equation, we get y = 2(3) + 6 = 12.
## The Solution
The solution to the system of linear equations is (3, 12).
Conclusion
In this article, we solved a system of two linear equations using graph paper or graphing technology. We graphed the two equations on a coordinate plane and found the point of intersection between the two graphs. The solution to the system of linear equations is (3, 12).
Answer
The solution to the system of linear equations is (3, 12).
Discussion
What is your favorite method for solving systems of linear equations? Do you prefer graphing, substitution, or elimination? Let us know in the comments below!
Related Topics
- Solving Systems of Linear Equations by Substitution
- Solving Systems of Linear Equations by Elimination
- Graphing Linear Equations
- Systems of Linear Equations with No Solutions
- Systems of Linear Equations with Infinite Solutions
Solving a System of Linear Equations: Q&A =============================================
Introduction
In our previous article, we solved a system of two linear equations using graph paper or graphing technology. In this article, we will answer some frequently asked questions about solving systems of linear equations.
Q: What is a system of linear equations?
A system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system.
Q: How do I know if a system of linear equations has a solution?
A system of linear equations has a solution if the two lines intersect at a single point. If the lines are parallel, the system has no solution. If the lines are the same, the system has infinite solutions.
Q: What is the difference between a system of linear equations with no solution and a system with infinite solutions?
A system of linear equations with no solution is a system where the two lines are parallel and never intersect. A system of linear equations with infinite solutions is a system where the two lines are the same and intersect at all points.
Q: How do I solve a system of linear equations using substitution?
To solve a system of linear equations using substitution, you can solve one of the equations for one of the variables and then substitute that expression into the other equation. This will give you a new equation with one variable, which you can then solve.
Q: How do I solve a system of linear equations using elimination?
To solve a system of linear equations using elimination, you can add or subtract the two equations to eliminate one of the variables. This will give you a new equation with one variable, which you can then solve.
Q: What is the point of intersection between two lines?
The point of intersection between two lines is the point where the two lines cross. This is the solution to the system of linear equations.
Q: How do I find the point of intersection between two lines?
To find the point of intersection between two lines, you can set the two equations equal to each other and solve for x. Then, you can substitute the value of x into one of the original equations to find the value of y.
Q: What is the solution to a system of linear equations?
The solution to a system of linear equations is the point where the two lines intersect. This is the point that satisfies both equations in the system.
Q: Can a system of linear equations have more than one solution?
No, a system of linear equations can only have one solution. If a system has infinite solutions, it means that the two lines are the same and intersect at all points.
Q: Can a system of linear equations have no solution?
Yes, a system of linear equations can have no solution. This means that the two lines are parallel and never intersect.
Conclusion
In this article, we answered some frequently asked questions about solving systems of linear equations. We discussed the difference between a system of linear equations with no solution and a system with infinite solutions, and we explained how to solve a system of linear equations using substitution and elimination.
Related Topics
- Solving Systems of Linear Equations by Substitution
- Solving Systems of Linear Equations by Elimination
- Graphing Linear Equations
- Systems of Linear Equations with No Solutions
- Systems of Linear Equations with Infinite Solutions
Discussion
Do you have any questions about solving systems of linear equations? Let us know in the comments below!
Additional Resources
- Khan Academy: Solving Systems of Linear Equations
- Mathway: Solving Systems of Linear Equations
- Wolfram Alpha: Solving Systems of Linear Equations