Solve The Pair Of Simultaneous Equations: $x+y=3$ And $2x-3y=16$A. $x=2$ And $y=1$B. $x=-2$ And $y=5$C. $x=5$ And $y=2$D. $x=5$ And $y=-2$

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Introduction

Simultaneous equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a pair of simultaneous equations, specifically the equations $x+y=3$ and $2x-3y=16$. We will use various methods to solve these equations and provide step-by-step solutions to help readers understand the process.

Understanding the Equations

Before we dive into solving the equations, let's take a closer look at what they represent. The first equation, $x+y=3$, represents a linear equation in two variables, $x$ and $y$. This equation states that the sum of $x$ and $y$ is equal to 3. The second equation, $2x-3y=16$, is also a linear equation in two variables, $x$ and $y$. This equation states that twice the value of $x$ minus three times the value of $y$ is equal to 16.

Method 1: Substitution Method

One way to solve simultaneous equations is by using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation. Let's use this method to solve the given equations.

Step 1: Solve the First Equation for $y$

We can start by solving the first equation for $y$. We can do this by isolating $y$ on one side of the equation. To do this, we can subtract $x$ from both sides of the equation, which gives us:

$$y = 3 - x$$

Step 2: Substitute the Expression for $y$ into the Second Equation

Now that we have an expression for $y$, we can substitute this expression into the second equation. We can do this by replacing $y$ with $3-x$ in the second equation. This gives us:

$$2x - 3(3-x) = 16$$

Step 3: Simplify the Equation

We can simplify the equation by distributing the negative 3 to the terms inside the parentheses. This gives us:

$$2x - 9 + 3x = 16$$

Step 4: Combine Like Terms

We can combine like terms by adding $2x$ and $3x$ to get $5x$. This gives us:

$$5x - 9 = 16$$

Step 5: Add 9 to Both Sides

We can add 9 to both sides of the equation to get:

$$5x = 25$$

Step 6: Divide Both Sides by 5

We can divide both sides of the equation by 5 to get:

$$x = 5$$

Step 7: Find the Value of $y$

Now that we have the value of $x$, we can find the value of $y$ by substituting $x$ into the expression for $y$. We can do this by replacing $x$ with 5 in the expression $y = 3 - x$. This gives us:

$$y = 3 - 5$$

$$y = -2$$

Method 2: Elimination Method

Another way to solve simultaneous 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. Let's use this method to solve the given equations.

Step 1: Multiply the First Equation by 3

We can start by multiplying the first equation by 3. This gives us:

$$3x + 3y = 9$$

Step 2: Multiply the Second Equation by 1

We can multiply the second equation by 1, which gives us:

$$2x - 3y = 16$$

Step 3: Add Both Equations

We can add both equations to eliminate the variable $y$. This gives us:

$$(3x + 3y) + (2x - 3y) = 9 + 16$$

$$5x = 25$$

Step 4: Divide Both Sides by 5

We can divide both sides of the equation by 5 to get:

$$x = 5$$

Step 5: Find the Value of $y$

Now that we have the value of $x$, we can find the value of $y$ by substituting $x$ into one of the original equations. We can do this by replacing $x$ with 5 in the first equation. This gives us:

$$5 + y = 3$$

$$y = -2$$

Conclusion

In this article, we have solved the pair of simultaneous equations $x+y=3$ and $2x-3y=16$ using two different methods: the substitution method and the elimination method. We have shown that both methods can be used to solve these equations and have provided step-by-step solutions to help readers understand the process. The solutions to the equations are $x=5$ and $y=-2$.

Final Answer

The final answer is: $\boxed{D}$

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Q&A: Solving Simultaneous Equations

Q: What are simultaneous equations?

A: Simultaneous equations are a pair of equations that involve two or more variables and are equal to the same value. In this case, we have two equations: $x+y=3$ and $2x-3y=16$.

Q: Why do we need to solve simultaneous equations?

A: Solving simultaneous equations is important in mathematics and real-life applications. It helps us to find the values of variables that satisfy both equations simultaneously.

Q: What are the different methods to solve simultaneous equations?

A: There are two main methods to solve simultaneous equations: the substitution method and the elimination method. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves multiplying both equations by necessary multiples such that the coefficients of $y$'s in both equations are the same.

Q: How do I choose which method to use?

A: The choice of method depends on the equations and the variables involved. If one equation is already solved for one variable, the substitution method may be easier to use. If the coefficients of $y$'s in both equations are the same, the elimination method may be easier to use.

Q: What if I get stuck while solving simultaneous equations?

A: If you get stuck while solving simultaneous equations, try to simplify the equations by combining like terms or multiplying both equations by necessary multiples. You can also try to use a different method or seek help from a teacher or tutor.

Q: Can I use technology to solve simultaneous equations?

A: Yes, you can use technology such as calculators or computer software to solve simultaneous equations. However, it's still important to understand the methods and steps involved in solving simultaneous equations.

Q: How do I check my answer?

A: To check your answer, substitute the values of $x$ and $y$ back into both original equations. If the equations are true, then your answer is correct.

Q: What if I get a different answer using a different method?

A: If you get a different answer using a different method, it's possible that one of the methods is incorrect. Double-check your work and make sure that you're using the correct method.

Q: Can I use simultaneous equations to solve real-life problems?

A: Yes, simultaneous equations can be used to solve real-life problems such as finance, engineering, and science. For example, you can use simultaneous equations to model the relationship between two variables and find the values of those variables.

Q: What are some common mistakes to avoid when solving simultaneous equations?

A: Some common mistakes to avoid when solving simultaneous equations include:

• Not checking your work
• Not using the correct method
• Not simplifying the equations
• Not substituting the values of $x$ and $y$ back into both original equations

Q: How can I practice solving simultaneous equations?

A: You can practice solving simultaneous equations by working on problems and exercises in your textbook or online resources. You can also try to create your own problems and solve them using different methods.

Conclusion

Solving simultaneous equations is an important skill in mathematics and real-life applications. By understanding the methods and steps involved in solving simultaneous equations, you can solve a wide range of problems and make informed decisions. Remember to check your work, use the correct method, and practice regularly to become proficient in solving simultaneous equations.

Final Answer

The final answer is: $\boxed{D}$