Solve The Following System Of Equations:${ \begin{array}{l} -3x - 8y = 20 \ -5x + Y = 19 \end{array} }$
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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 involves finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables.
The System of Equations
The system of equations we will be solving is:
{ \begin{array}{l} -3x - 8y = 20 \\ -5x + y = 19 \end{array} \}
Method 1: Substitution Method
One way to solve this system of equations is by using the substitution method. This method involves solving one of the equations for one variable and then substituting that expression into the other equation.
Step 1: Solve the Second Equation for y
We can solve the second equation for y by adding 5x to both sides of the equation and then subtracting 19 from both sides.
{ y = -5x + 19 \}
Step 2: Substitute the Expression for y into the First Equation
Now that we have an expression for y, we can substitute it into the first equation.
{ -3x - 8(-5x + 19) = 20 \}
Step 3: Simplify the Equation
Next, we can simplify the equation by distributing the -8 to the terms inside the parentheses.
{ -3x + 40x - 152 = 20 \}
Step 4: Combine Like Terms
Now, we can combine like terms by adding -3x and 40x.
{ 37x - 152 = 20 \}
Step 5: Add 152 to Both Sides
Next, we can add 152 to both sides of the equation to isolate the term with the variable.
{ 37x = 172 \}
Step 6: Divide Both Sides by 37
Finally, we can divide both sides of the equation by 37 to solve for x.
{ x = \frac{172}{37} \}
Step 7: Find the Value of y
Now that we have the value of x, we can substitute it into the expression for y that we found in Step 1.
{ y = -5\left(\frac{172}{37}\right) + 19 \}
Step 8: Simplify the Expression for y
Next, we can simplify the expression for y by multiplying -5 by 172 and then dividing by 37.
{ y = -\frac{860}{37} + 19 \}
Step 9: Add 19 to Both Sides
Now, we can add 19 to both sides of the equation to simplify the expression for y.
{ y = -\frac{860}{37} + \frac{703}{37} \}
Step 10: Combine Like Terms
Finally, we can combine like terms by adding -860/37 and 703/37.
{ y = -\frac{157}{37} \}
Method 2: Elimination Method
Another way to solve this system of equations is by using the elimination method. This method involves multiplying both equations by necessary multiples such that the coefficients of y's in both equations are the same.
Step 1: Multiply the First Equation by 1 and the Second Equation by 8
We can multiply the first equation by 1 and the second equation by 8 to make the coefficients of y's in both equations the same.
{ -3x - 8y = 20 \}
{ -40x + 8y = 152 \}
Step 2: Add Both Equations
Now that the coefficients of y's in both equations are the same, we can add both equations to eliminate the variable y.
{ -43x = 172 \}
Step 3: Divide Both Sides by -43
Next, we can divide both sides of the equation by -43 to solve for x.
{ x = -\frac{172}{43} \}
Step 4: Substitute the Value of x into One of the Original Equations
Now that we have the value of x, we can substitute it into one of the original equations to solve for y.
{ -3\left(-\frac{172}{43}\right) - 8y = 20 \}
Step 5: Simplify the Equation
Next, we can simplify the equation by multiplying -3 by -172 and then dividing by 43.
{ \frac{516}{43} - 8y = 20 \}
Step 6: Subtract 516/43 from Both Sides
Now, we can subtract 516/43 from both sides of the equation to isolate the term with the variable y.
{ -8y = 20 - \frac{516}{43} \}
Step 7: Simplify the Right-Hand Side
Next, we can simplify the right-hand side of the equation by finding a common denominator and then subtracting the fractions.
{ -8y = \frac{860}{43} - \frac{516}{43} \}
Step 8: Combine Like Terms
Now, we can combine like terms by subtracting 516/43 from 860/43.
{ -8y = \frac{344}{43} \}
Step 9: Divide Both Sides by -8
Finally, we can divide both sides of the equation by -8 to solve for y.
{ y = -\frac{344}{344} \}
Step 10: Simplify the Expression for y
Now, we can simplify the expression for y by canceling out the common factor of 344.
{ y = -\frac{1}{1} \}
Conclusion
In this article, we have solved a system of two linear equations with two variables using two different methods: the substitution method and the elimination method. Both methods involve finding the values of the variables that satisfy all the equations in the system. The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation, while the elimination method involves multiplying both equations by necessary multiples such that the coefficients of y's in both equations are the same. By following these steps, we can solve systems of linear equations and find the values of the variables that satisfy all the equations in the system.
Final Answer
The final answer is:
{ x = -\frac{172}{43} \}
{
y = -\frac{1}{1}
\}$<br/>
# **Solving a System of Linear Equations: A Q&A Guide**
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In our previous article, we solved a system of two linear equations with two variables using two different methods: the substitution method and the elimination method. In this article, we will provide a Q&A guide to help you understand the concepts and methods involved in solving systems of linear equations. A: 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 involves finding the values of the variables that satisfy all the equations in the system. A: The two main methods for solving systems of linear equations are the substitution method and the elimination method. The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation, while the elimination method involves multiplying both equations by necessary multiples such that the coefficients of y's in both equations are the same. A: The choice of method depends on the specific system of equations and the variables involved. If the coefficients of one variable are the same in both equations, the elimination method may be easier to use. If the coefficients of one variable are different in both equations, the substitution method may be easier to use. A: Some common mistakes to avoid when solving systems of linear equations include: A: To check if a solution is extraneous, plug the values of the variables into both equations and check if they are true. If the solution does not satisfy both equations, it is an extraneous solution and should be discarded. A: Solving systems of linear equations has many real-world applications, including: A: Yes, you can use a calculator to solve systems of linear equations. Many calculators have built-in functions for solving systems of linear equations, such as the "solve" function. A: To graph a system of linear equations, plot the two lines on a coordinate plane and find the point of intersection. The point of intersection is the solution to the system of equations. A: Some common types of systems of linear equations include: In this article, we have provided a Q&A guide to help you understand the concepts and methods involved in solving systems of linear equations. We have covered topics such as the two main methods for solving systems of linear equations, common mistakes to avoid, and real-world applications of solving systems of linear equations. By following these tips and guidelines, you can become proficient in solving systems of linear equations and apply this knowledge to a wide range of problems. The final answer is:Introduction
Q: What is a system of linear equations?
Q: What are the two main methods for solving systems of linear equations?
Q: How do I choose which method to use?
Q: What are some common mistakes to avoid when solving systems of linear equations?
Q: How do I check if a solution is extraneous?
Q: What are some real-world applications of solving systems of linear equations?
Q: Can I use a calculator to solve systems of linear equations?
Q: How do I graph a system of linear equations?
Q: What are some common types of systems of linear equations?
Conclusion
Final Answer