Solve This System Of Equations Using Substitution:$ \begin{array}{l} y = 2x - 4 \ x + 2y = 10 \end{array} }$Step 1 Substitute For { Y$ $ In The Second Equation. What Is The Resulting Equation?$[ X + 2(2x - 4) = 10

Mar 10, 2025 by ADMIN 215 views

**Solving Systems of Equations using Substitution**

Introduction

Solving systems of equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. In this article, we will focus on solving a system of linear equations using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation to solve for the remaining variable.

Step 1: Substitute for y in the Second Equation

The given system of equations is:

{ \begin{array}{l} y = 2x - 4 \\ x + 2y = 10 \end{array} \}

To solve this system using substitution, we will first substitute the expression for y from the first equation into the second equation. This will give us an equation in terms of x, which we can then solve to find the value of x.

The first equation is:

y = 2x - 4

We will substitute this expression for y into the second equation:

x + 2y = 10

Substituting y = 2x - 4 into the second equation, we get:

x + 2(2x - 4) = 10

Simplifying the Equation

Now, let's simplify the equation by evaluating the expression inside the parentheses:

x + 2(2x - 4) = 10

Using the distributive property, we can rewrite the equation as:

x + 4x - 8 = 10

Combining like terms, we get:

5x - 8 = 10

Adding 8 to Both Sides

To isolate the term with x, we will add 8 to both sides of the equation:

5x - 8 + 8 = 10 + 8

This simplifies to:

5x = 18

Dividing Both Sides by 5

Finally, we will divide both sides of the equation by 5 to solve for x:

5x/5 = 18/5

This gives us:

x = 18/5

Conclusion

In this article, we used the substitution method to solve a system of linear equations. We substituted the expression for y from the first equation into the second equation and then simplified the resulting equation to solve for x. The final answer is x = 18/5. This value of x can then be used to find the corresponding value of y by substituting it into one of the original equations.

Discussion

The substitution method is a powerful tool for solving systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation to solve for the remaining variable. This method is particularly useful when one of the equations is already solved for one variable.

Example 2: Solving a System of Equations using Substitution

Let's consider another example of a system of equations:

{ \begin{array}{l} y = 3x + 2 \\ 2x + 5y = 12 \end{array} \}

The first equation is:

y = 3x + 2

We will substitute this expression for y into the second equation:

2x + 5y = 12

Substituting y = 3x + 2 into the second equation, we get:

2x + 5(3x + 2) = 12

Simplifying the Equation

Now, let's simplify the equation by evaluating the expression inside the parentheses:

2x + 5(3x + 2) = 12

Using the distributive property, we can rewrite the equation as:

2x + 15x + 10 = 12

Combining like terms, we get:

17x + 10 = 12

Subtracting 10 from Both Sides

To isolate the term with x, we will subtract 10 from both sides of the equation:

17x + 10 - 10 = 12 - 10

This simplifies to:

17x = 2

Dividing Both Sides by 17

Finally, we will divide both sides of the equation by 17 to solve for x:

17x/17 = 2/17

This gives us:

x = 2/17

Conclusion

In this article, we used the substitution method to solve a system of linear equations. We substituted the expression for y from the first equation into the second equation and then simplified the resulting equation to solve for x. The final answer is x = 2/17. This value of x can then be used to find the corresponding value of y by substituting it into one of the original equations.

Discussion

Tips and Tricks

Here are some tips and tricks to help you solve systems of equations using substitution:

Make sure to substitute the expression for y from the first equation into the second equation correctly.
Simplify the resulting equation by combining like terms and evaluating expressions inside parentheses.
Isolate the term with x by adding or subtracting the same value from both sides of the equation.
Divide both sides of the equation by the coefficient of x to solve for x.

Introduction

In our previous article, we discussed how to solve systems of equations using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation to solve for the remaining variable. In this article, we will answer some frequently asked questions about solving systems of equations using substitution.

Q: What is the substitution method?

A: The substitution method is a technique used to solve systems of equations by substituting the expression for one variable from one equation 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. This method is particularly useful when one of the equations is linear and the other equation is quadratic or has a more complex expression.

Q: How do I know which equation to substitute into?

A: You should substitute the expression for y from the first equation into the second equation. This is because the first equation is usually solved for y, and substituting it into the second equation will give you an equation in terms of x.

Q: What if I have a system of equations with two variables and two equations?

A: In this case, you can use the substitution method to solve for one variable and then substitute that expression into the other equation to solve for the remaining variable.

Q: Can I use the substitution method to solve systems of equations with more than two variables?

A: Yes, you can use the substitution method to solve systems of equations with more than two variables. However, this method can become more complex and may require additional steps to solve.

Q: What if I have a system of equations with fractions or decimals?

A: You can use the substitution method to solve systems of equations with fractions or decimals. However, you may need to simplify the resulting equation by combining like terms and evaluating expressions inside parentheses.

Q: Can I use the substitution method to solve systems of equations with absolute values or inequalities?

A: No, the substitution method is not suitable for solving systems of equations with absolute values or inequalities. In these cases, you may need to use other methods such as the graphical method or the algebraic method.

Q: How do I check my answer?

A: To check your answer, you should substitute the value of x back into one of the original equations to find the corresponding value of y. If the values of x and y satisfy both equations, then your answer is correct.

Q: What are some common mistakes to avoid when using the substitution method?

A: Some common mistakes to avoid when using the substitution method include:

Not simplifying the resulting equation by combining like terms and evaluating expressions inside parentheses.
Not isolating the term with x by adding or subtracting the same value from both sides of the equation.
Not checking the answer by substituting the value of x back into one of the original equations.

Conclusion

In this article, we answered some frequently asked questions about solving systems of equations using substitution. We discussed the substitution method, when to use it, and how to avoid common mistakes. By following these tips and tricks, you can master the substitution method and solve systems of equations with ease.

Tips and Tricks

Here are some additional tips and tricks to help you solve systems of equations using substitution:

Make sure to simplify the resulting equation by combining like terms and evaluating expressions inside parentheses.
Isolate the term with x by adding or subtracting the same value from both sides of the equation.
Check your answer by substituting the value of x back into one of the original equations.
Use the substitution method to solve systems of equations with fractions or decimals.
Avoid using the substitution method to solve systems of equations with absolute values or inequalities.

By following these tips and tricks, you can become a master of solving systems of equations using substitution.