Solve The System Of Equations By Substitution.${ \begin{cases} x = 2y + 11 \ 3x + 7y = -32 \end{cases} }$The Solution Of The System Is { X = \square$}$ And { Y = \square$}$.(Type Integers Or Simplified Fractions.)
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Introduction
In this article, we will learn how to solve a system of linear equations using the substitution method. This method involves solving one equation for a variable and then substituting that expression into the other equation to solve for the other variable. We will use a system of two linear equations in two variables to demonstrate this method.
The System of Equations
The system of equations we will be solving is:
{ \begin{cases} x = 2y + 11 \\ 3x + 7y = -32 \end{cases} \}
Step 1: Solve the First Equation for x
The first equation is already solved for x, so we can write it as:
x = 2y + 11
Step 2: Substitute the Expression for x into the Second Equation
Now, we will substitute the expression for x into the second equation:
3(2y + 11) + 7y = -32
Step 3: Expand and Simplify the Equation
Next, we will expand and simplify the equation:
6y + 33 + 7y = -32
Combine like terms:
13y + 33 = -32
Step 4: Isolate the Variable y
Now, we will isolate the variable y by subtracting 33 from both sides of the equation:
13y = -32 - 33
13y = -65
Step 5: Solve for y
Finally, we will solve for y by dividing both sides of the equation by 13:
y = -65/13
y = -5
Step 6: Substitute the Value of y into the First Equation
Now that we have the value of y, we can substitute it into the first equation to solve for x:
x = 2(-5) + 11
x = -10 + 11
x = 1
Conclusion
In this article, we learned how to solve a system of linear equations using the substitution method. We used a system of two linear equations in two variables to demonstrate this method. We solved for the variable y by isolating it in the second equation and then substituted the value of y into the first equation to solve for the variable x.
Example Use Case
The substitution method can be used to solve systems of linear equations in a variety of situations, such as:
- Solving systems of linear equations in two variables
- Solving systems of linear equations in three variables
- Solving systems of linear equations with fractions or decimals
- Solving systems of linear equations with negative numbers
Tips and Tricks
Here are some tips and tricks to keep in mind when using the substitution method:
- Make sure to solve one equation for a variable before substituting it into the other equation.
- Use the correct order of operations when simplifying the equation.
- Check your work by plugging the values of x and y back into the original equations.
Common Mistakes
Here are some common mistakes to avoid when using the substitution method:
- Failing to solve one equation for a variable before substituting it into the other equation.
- Not using the correct order of operations when simplifying the equation.
- Not checking your work by plugging the values of x and y back into the original equations.
Real-World Applications
The substitution method has many real-world applications, such as:
- Solving systems of linear equations in physics and engineering
- Solving systems of linear equations in economics and finance
- Solving systems of linear equations in computer science and programming
Conclusion
In conclusion, the substitution method is a powerful tool for solving systems of linear equations. By following the steps outlined in this article, you can use the substitution method to solve systems of linear equations in a variety of situations. Remember to check your work and avoid common mistakes to ensure that you get the correct solution.
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Introduction
In our previous article, we learned how to solve a system of linear equations using the substitution method. In this article, we will answer some frequently asked questions about the substitution method and provide additional examples to help you understand the concept better.
Q&A
Q: What is the substitution method?
A: The substitution method is a technique used to solve systems of linear equations by substituting the expression for one variable into the other equation.
Q: When should I use the substitution method?
A: You should use the substitution method when one of the equations is already solved for one variable, or when you can easily solve one equation for one variable.
Q: How do I know which equation to solve for first?
A: You should solve the equation that is easiest to solve for one variable first. If one equation is already solved for one variable, you can use that equation as the first equation.
Q: What if I have a system of linear equations with fractions or decimals?
A: You can still use the substitution method to solve a system of linear equations with fractions or decimals. Just make sure to simplify the equation correctly.
Q: What if I have a system of linear equations with negative numbers?
A: You can still use the substitution method to solve a system of linear equations with negative numbers. Just make sure to simplify the equation correctly.
Q: How do I check my work?
A: You should check your work by plugging the values of x and y back into the original equations to make sure they are true.
Q: What if I get a system of linear equations with no solution?
A: If you get a system of linear equations with no solution, it means that the two equations are inconsistent and there is no solution.
Q: What if I get a system of linear equations with infinitely many solutions?
A: If you get a system of linear equations with infinitely many solutions, it means that the two equations are dependent and there are infinitely many solutions.
Examples
Example 1: Solving a System of Linear Equations with Fractions
Solve the system of linear equations:
{ \begin{cases} x = \frac{1}{2}y + 3 \\ 2x + 3y = 5 \end{cases} \}
Solution
First, we will solve the first equation for x:
x = (1/2)y + 3
Next, we will substitute the expression for x into the second equation:
2((1/2)y + 3) + 3y = 5
Simplify the equation:
y + 6 + 3y = 5
Combine like terms:
4y + 6 = 5
Subtract 6 from both sides:
4y = -1
Divide both sides by 4:
y = -1/4
Now that we have the value of y, we can substitute it into the first equation to solve for x:
x = (1/2)(-1/4) + 3
x = -1/8 + 3
x = 23/8
Example 2: Solving a System of Linear Equations with Negative Numbers
Solve the system of linear equations:
{ \begin{cases} x = -2y - 4 \\ 3x + 2y = -10 \end{cases} \}
Solution
First, we will solve the first equation for x:
x = -2y - 4
Next, we will substitute the expression for x into the second equation:
3(-2y - 4) + 2y = -10
Simplify the equation:
-6y - 12 + 2y = -10
Combine like terms:
-4y - 12 = -10
Add 12 to both sides:
-4y = 2
Divide both sides by -4:
y = -1/2
Now that we have the value of y, we can substitute it into the first equation to solve for x:
x = -2(-1/2) - 4
x = 1 - 4
x = -3
Conclusion
In this article, we answered some frequently asked questions about the substitution method and provided additional examples to help you understand the concept better. Remember to check your work and avoid common mistakes to ensure that you get the correct solution.
Tips and Tricks
Here are some tips and tricks to keep in mind when using the substitution method:
- Make sure to solve one equation for a variable before substituting it into the other equation.
- Use the correct order of operations when simplifying the equation.
- Check your work by plugging the values of x and y back into the original equations.
Common Mistakes
Here are some common mistakes to avoid when using the substitution method:
- Failing to solve one equation for a variable before substituting it into the other equation.
- Not using the correct order of operations when simplifying the equation.
- Not checking your work by plugging the values of x and y back into the original equations.
Real-World Applications
The substitution method has many real-world applications, such as:
- Solving systems of linear equations in physics and engineering
- Solving systems of linear equations in economics and finance
- Solving systems of linear equations in computer science and programming