Solve The System Of Equations By The Substitution Method.(i) S − T = 3 S - T = 3 S − T = 3 (ii) S 3 + T 2 = 6 \frac{s}{3} + \frac{t}{2} = 6 3 S ​ + 2 T ​ = 6

Introduction

Solving a system of equations is a fundamental concept in mathematics, and it is essential to understand various methods to solve them. The substitution method is one of the most commonly used techniques to solve a system of equations. In this article, we will learn how to solve a system of equations using the substitution method.

What is the Substitution Method?

The substitution method is a technique used to solve a system of equations by substituting the expression of one variable from one equation into the other equation. This method is particularly useful when one of the equations is linear and the other is quadratic or when one of the variables is easily expressed in terms of the other variable.

Step 1: Identify the Equations

To solve a system of equations using the substitution method, we need to identify the two equations. In this case, we have two equations:

(i) $s - t = 3$

(ii) $\frac{s}{3} + \frac{t}{2} = 6$

Step 2: Solve One Equation for One Variable

We will start by solving equation (i) for the variable $s$. We can add $t$ to both sides of the equation to get:

$s = t + 3$

Step 3: Substitute the Expression into the Other Equation

Now, we will substitute the expression of $s$ from equation (i) into equation (ii). We can replace $s$ with $t + 3$ in equation (ii):

$\frac{t + 3}{3} + \frac{t}{2} = 6$

Step 4: Simplify the Equation

To simplify the equation, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is 6. This will eliminate the fractions:

$2(t + 3) + 3t = 36$

Step 5: Expand and Simplify the Equation

We can expand and simplify the equation by distributing the 2 and combining like terms:

$2t + 6 + 3t = 36$

$5t + 6 = 36$

Step 6: Solve for the Variable

Now, we can solve for the variable $t$ by subtracting 6 from both sides of the equation and then dividing both sides by 5:

$5t = 30$

$t = 6$

Step 7: Find the Value of the Other Variable

Now that we have the value of $t$, we can find the value of $s$ by substituting $t$ into one of the original equations. We will use equation (i):

$s - t = 3$

$s - 6 = 3$

$s = 9$

Conclusion

In this article, we learned how to solve a system of equations using the substitution method. We started by identifying the two equations, solving one equation for one variable, substituting the expression into the other equation, simplifying the equation, expanding and simplifying the equation, solving for the variable, and finding the value of the other variable. By following these steps, we can solve a system of equations using the substitution method.

Example 1: Solve the System of Equations

Solve the system of equations using the substitution method:

(i) $x + y = 4$

(ii) $2x - 3y = -2$

Step 1: Solve One Equation for One Variable

We will start by solving equation (i) for the variable $x$. We can subtract $y$ from both sides of the equation to get:

$x = 4 - y$

Step 2: Substitute the Expression into the Other Equation

Now, we will substitute the expression of $x$ from equation (i) into equation (ii). We can replace $x$ with $4 - y$ in equation (ii):

$2(4 - y) - 3y = -2$

Step 3: Simplify the Equation

To simplify the equation, we can distribute the 2 and combine like terms:

$8 - 2y - 3y = -2$

$8 - 5y = -2$

Step 4: Solve for the Variable

Now, we can solve for the variable $y$ by subtracting 8 from both sides of the equation and then dividing both sides by -5:

$-5y = -10$

$y = 2$

Step 5: Find the Value of the Other Variable

Now that we have the value of $y$, we can find the value of $x$ by substituting $y$ into one of the original equations. We will use equation (i):

$x + y = 4$

$x + 2 = 4$

$x = 2$

Conclusion

In this example, we solved a system of equations using the substitution method. We started by identifying the two equations, solving one equation for one variable, substituting the expression into the other equation, simplifying the equation, solving for the variable, and finding the value of the other variable.

Example 2: Solve the System of Equations

Solve the system of equations using the substitution method:

(i) $x - 2y = 3$

(ii) $x + 3y = 5$

Step 1: Solve One Equation for One Variable

We will start by solving equation (i) for the variable $x$. We can add $2y$ to both sides of the equation to get:

$x = 2y + 3$

Step 2: Substitute the Expression into the Other Equation

Now, we will substitute the expression of $x$ from equation (i) into equation (ii). We can replace $x$ with $2y + 3$ in equation (ii):

$2y + 3 + 3y = 5$

Step 3: Simplify the Equation

To simplify the equation, we can combine like terms:

$5y + 3 = 5$

Step 4: Solve for the Variable

Now, we can solve for the variable $y$ by subtracting 3 from both sides of the equation and then dividing both sides by 5:

$5y = 2$

$y = \frac{2}{5}$

Step 5: Find the Value of the Other Variable

Now that we have the value of $y$, we can find the value of $x$ by substituting $y$ into one of the original equations. We will use equation (i):

$x - 2y = 3$

$x - 2(\frac{2}{5}) = 3$

$x - \frac{4}{5} = 3$

$x = \frac{19}{5}$

Conclusion

In this example, we solved a system of equations using the substitution method. We started by identifying the two equations, solving one equation for one variable, substituting the expression into the other equation, simplifying the equation, solving for the variable, and finding the value of the other variable.

Advantages of the Substitution Method

The substitution method has several advantages. It is particularly useful when one of the equations is linear and the other is quadratic or when one of the variables is easily expressed in terms of the other variable. This method also helps to eliminate one of the variables, making it easier to solve for the other variable.

Disadvantages of the Substitution Method

The substitution method also has some disadvantages. It can be time-consuming and tedious, especially when the equations are complex. Additionally, this method may not be suitable for all types of equations, such as non-linear equations.

Conclusion

In conclusion, the substitution method is a powerful technique used to solve a system of equations. It involves solving one equation for one variable, substituting the expression into the other equation, simplifying the equation, solving for the variable, and finding the value of the other variable. By following these steps, we can solve a system of equations using the substitution method. This method has several advantages, including the ability to eliminate one of the variables and make it easier to solve for the other variable. However, it also has some disadvantages, such as being time-consuming and tedious.

Introduction

Solving a system of equations is a fundamental concept in mathematics, and it is essential to understand various methods to solve them. The substitution method is one of the most commonly used techniques to solve a system of equations. In this article, we will answer some frequently asked questions about the substitution method.

Q: What is the substitution method?

A: The substitution method is a technique used to solve a system of equations by substituting the expression of 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 linear and the other is quadratic or when one of the variables is easily expressed in terms of the other variable.

Q: How do I start solving a system of equations using the substitution method?

A: To start solving a system of equations using the substitution method, you need to identify the two equations and solve one equation for one variable.

Q: What if I have two linear equations?

A: If you have two linear equations, you can use the substitution method by solving one equation for one variable and substituting the expression into the other equation.

Q: What if I have a non-linear equation?

A: If you have a non-linear equation, you may not be able to use the substitution method. In this case, you may need to use other methods, such as the elimination method or the graphing method.

Q: How do I simplify the equation after substituting the expression?

A: To simplify the equation after substituting the expression, you can combine like terms and eliminate any fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

Q: What if I get stuck during the substitution method?

A: If you get stuck during the substitution method, you can try to simplify the equation further or use a different method, such as the elimination method or the graphing method.

Q: Can I use the substitution method to solve a system of three or more equations?

A: Yes, you can use the substitution method to solve a system of three or more equations. However, it may be more complicated and time-consuming.

Q: What are the advantages of the substitution method?

A: The advantages of the substitution method include the ability to eliminate one of the variables, making it easier to solve for the other variable, and the ability to use it with linear and quadratic equations.

Q: What are the disadvantages of the substitution method?

A: The disadvantages of the substitution method include the potential for it to be time-consuming and tedious, especially with complex equations, and the possibility of getting stuck during the process.

Q: Can I use the substitution method to solve a system of equations with fractions?

A: Yes, you can use the substitution method to solve a system of equations with fractions. However, you may need to multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.

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

A: If you have a system of equations with decimals, you can use the substitution method by converting the decimals to fractions and then following the same steps as before.

Q: Can I use the substitution method to solve a system of equations with negative numbers?

A: Yes, you can use the substitution method to solve a system of equations with negative numbers. However, you may need to be careful when simplifying the equation and solving for the variables.

Conclusion

In conclusion, the substitution method is a powerful technique used to solve a system of equations. It involves solving one equation for one variable, substituting the expression into the other equation, simplifying the equation, solving for the variable, and finding the value of the other variable. By following these steps and understanding the advantages and disadvantages of the substitution method, you can solve a system of equations using this technique.

Example 1: Solve the System of Equations

Solve the system of equations using the substitution method:

(i) $x + y = 4$

(ii) $2x - 3y = -2$

Step 1: Solve One Equation for One Variable

We will start by solving equation (i) for the variable $x$. We can subtract $y$ from both sides of the equation to get:

$x = 4 - y$

Step 2: Substitute the Expression into the Other Equation

Now, we will substitute the expression of $x$ from equation (i) into equation (ii). We can replace $x$ with $4 - y$ in equation (ii):

$2(4 - y) - 3y = -2$

Step 3: Simplify the Equation

To simplify the equation, we can distribute the 2 and combine like terms:

$8 - 2y - 3y = -2$

$8 - 5y = -2$

Step 4: Solve for the Variable

Now, we can solve for the variable $y$ by subtracting 8 from both sides of the equation and then dividing both sides by -5:

$-5y = -10$

$y = 2$

Step 5: Find the Value of the Other Variable

Now that we have the value of $y$, we can find the value of $x$ by substituting $y$ into one of the original equations. We will use equation (i):

$x + y = 4$

$x + 2 = 4$

$x = 2$

Conclusion

In this example, we solved a system of equations using the substitution method. We started by identifying the two equations, solving one equation for one variable, substituting the expression into the other equation, simplifying the equation, solving for the variable, and finding the value of the other variable.

Example 2: Solve the System of Equations

Solve the system of equations using the substitution method:

(i) $x - 2y = 3$

(ii) $x + 3y = 5$

Step 1: Solve One Equation for One Variable

We will start by solving equation (i) for the variable $x$. We can add $2y$ to both sides of the equation to get:

$x = 2y + 3$

Step 2: Substitute the Expression into the Other Equation

Now, we will substitute the expression of $x$ from equation (i) into equation (ii). We can replace $x$ with $2y + 3$ in equation (ii):

$2y + 3 + 3y = 5$

Step 3: Simplify the Equation

To simplify the equation, we can combine like terms:

$5y + 3 = 5$

Step 4: Solve for the Variable

Now, we can solve for the variable $y$ by subtracting 3 from both sides of the equation and then dividing both sides by 5:

$5y = 2$

$y = \frac{2}{5}$

Step 5: Find the Value of the Other Variable

Now that we have the value of $y$, we can find the value of $x$ by substituting $y$ into one of the original equations. We will use equation (i):

$x - 2y = 3$

$x - 2(\frac{2}{5}) = 3$

$x - \frac{4}{5} = 3$

$x = \frac{19}{5}$

Conclusion

In this example, we solved a system of equations using the substitution method. We started by identifying the two equations, solving one equation for one variable, substituting the expression into the other equation, simplifying the equation, solving for the variable, and finding the value of the other variable.

Conclusion

In conclusion, the substitution method is a powerful technique used to solve a system of equations. It involves solving one equation for one variable, substituting the expression into the other equation, simplifying the equation, solving for the variable, and finding the value of the other variable. By following these steps and understanding the advantages and disadvantages of the substitution method, you can solve a system of equations using this technique.