Verify If \[$\left(\frac{5}{3},-\frac{2}{3}\right)\$\] Is A Solution To The System Of Equations:$\[ \begin{align*} y &= \frac{1}{2} X - \frac{3}{2} \\ y &= -x + 1 \end{align*} \\]Is The Ordered Pair A Solution To The System?A. Yes,

Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. To verify if a given ordered pair is a solution to a system of equations, we need to substitute the values of the ordered pair into each equation and check if the equation holds true. In this article, we will verify if the ordered pair $\left(\frac{5}{3},-\frac{2}{3}\right)$ is a solution to the system of equations:

$$\begin{align*} y &= \frac{1}{2} x - \frac{3}{2} \\ y &= -x + 1 \end{align*}$$

Understanding the System of Equations

The given system of equations consists of two linear equations in the form of $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The first equation is $y = \frac{1}{2} x - \frac{3}{2}$, which has a slope of $\frac{1}{2}$ and a y-intercept of $-\frac{3}{2}$. The second equation is $y = -x + 1$, which has a slope of $-1$ and a y-intercept of $1$.

Verifying the Solution

To verify if the ordered pair $\left(\frac{5}{3},-\frac{2}{3}\right)$ is a solution to the system of equations, we need to substitute the values of $x$ and $y$ into each equation and check if the equation holds true.

Substituting into the First Equation

Let's substitute $x = \frac{5}{3}$ and $y = -\frac{2}{3}$ into the first equation:

$$y = \frac{1}{2} x - \frac{3}{2}$$

$$-\frac{2}{3} = \frac{1}{2} \left(\frac{5}{3}\right) - \frac{3}{2}$$

To simplify the equation, we can multiply both sides by $6$ to eliminate the fractions:

$$-4 = 5 - 9$$

$$-4 = -4$$

The equation holds true, which means that the ordered pair $\left(\frac{5}{3},-\frac{2}{3}\right)$ satisfies the first equation.

Substituting into the Second Equation

Now, let's substitute $x = \frac{5}{3}$ and $y = -\frac{2}{3}$ into the second equation:

$$y = -x + 1$$

$$-\frac{2}{3} = -\left(\frac{5}{3}\right) + 1$$

To simplify the equation, we can multiply both sides by $3$ to eliminate the fractions:

$$-2 = -5 + 3$$

$$-2 = -2$$

The equation holds true, which means that the ordered pair $\left(\frac{5}{3},-\frac{2}{3}\right)$ satisfies the second equation.

Conclusion

Since the ordered pair $\left(\frac{5}{3},-\frac{2}{3}\right)$ satisfies both equations in the system, we can conclude that it is a solution to the system of equations.

Discussion

The process of verifying a solution to a system of equations involves substituting the values of the ordered pair into each equation and checking if the equation holds true. In this case, we used the given ordered pair $\left(\frac{5}{3},-\frac{2}{3}\right)$ and substituted it into both equations in the system. By simplifying the equations and checking if they hold true, we were able to verify that the ordered pair is indeed a solution to the system.

Example Problems

Here are a few example problems that involve verifying solutions to a system of equations:

1. Verify if the ordered pair $\left(2,3\right)$ is a solution to the system of equations:

$$\begin{align*} y &= 2x - 1 \\ y &= x + 2 \end{align*}$$

2. Verify if the ordered pair $\left(-1,2\right)$ is a solution to the system of equations:

$$\begin{align*} y &= -x + 3 \\ y &= 2x - 1 \end{align*}$$

3. Verify if the ordered pair $\left(0,0\right)$ is a solution to the system of equations:

$$\begin{align*} y &= 2x - 1 \\ y &= -x + 1 \end{align*}$$

Solutions

1. To verify if the ordered pair $\left(2,3\right)$ is a solution to the system of equations, we need to substitute the values of $x$ and $y$ into each equation and check if the equation holds true.

Substituting into the first equation:

$$y = 2x - 1$$

$$3 = 2(2) - 1$$

$$3 = 3$$

The equation holds true, which means that the ordered pair $\left(2,3\right)$ satisfies the first equation.

Substituting into the second equation:

$$y = x + 2$$

$$3 = 2 + 2$$

$$3 = 4$$

The equation does not hold true, which means that the ordered pair $\left(2,3\right)$ does not satisfy the second equation.

Therefore, the ordered pair $\left(2,3\right)$ is not a solution to the system of equations.

2. To verify if the ordered pair $\left(-1,2\right)$ is a solution to the system of equations, we need to substitute the values of $x$ and $y$ into each equation and check if the equation holds true.

Substituting into the first equation:

$$y = -x + 3$$

$$2 = -(-1) + 3$$

$$2 = 4$$

The equation does not hold true, which means that the ordered pair $\left(-1,2\right)$ does not satisfy the first equation.

Substituting into the second equation:

$$y = 2x - 1$$

$$2 = 2(-1) - 1$$

$$2 = -3$$

The equation does not hold true, which means that the ordered pair $\left(-1,2\right)$ does not satisfy the second equation.

Therefore, the ordered pair $\left(-1,2\right)$ is not a solution to the system of equations.

3. To verify if the ordered pair $\left(0,0\right)$ is a solution to the system of equations, we need to substitute the values of $x$ and $y$ into each equation and check if the equation holds true.

Substituting into the first equation:

$$y = 2x - 1$$

$$0 = 2(0) - 1$$

$$0 = -1$$

The equation does not hold true, which means that the ordered pair $\left(0,0\right)$ does not satisfy the first equation.

Substituting into the second equation:

$$y = -x + 1$$

$$0 = -0 + 1$$

$$0 = 1$$

The equation does not hold true, which means that the ordered pair $\left(0,0\right)$ does not satisfy the second equation.

Therefore, the ordered pair $\left(0,0\right)$ is not a solution to the system of equations.

Conclusion

In this article, we verified if the ordered pair $\left(\frac{5}{3},-\frac{2}{3}\right)$ is a solution to the system of equations:

$$\begin{align*} y &= \frac{1}{2} x - \frac{3}{2} \\ y &= -x + 1 \end{align*}$$

Introduction

In our previous article, we discussed how to verify if a given ordered pair is a solution to a system of equations. We used the system of equations:

$$\begin{align*} y &= \frac{1}{2} x - \frac{3}{2} \\ y &= -x + 1 \end{align*}$$

and verified that the ordered pair $\left(\frac{5}{3},-\frac{2}{3}\right)$ is a solution to the system. In this article, we will answer some frequently asked questions about verifying solutions to a system of equations.

Q&A

Q: What is the first step in verifying a solution to a system of equations?

A: The first step in verifying a solution to a system of equations is to substitute the values of the ordered pair into each equation in the system.

Q: How do I know if an ordered pair is a solution to a system of equations?

A: An ordered pair is a solution to a system of equations if it satisfies both equations in the system. To verify this, you need to substitute the values of the ordered pair into each equation and check if the equation holds true.

Q: What if the ordered pair satisfies one equation but not the other?

A: If the ordered pair satisfies one equation but not the other, then it is not a solution to the system of equations.

Q: Can an ordered pair be a solution to a system of equations if it satisfies only one of the equations?

A: No, an ordered pair cannot be a solution to a system of equations if it satisfies only one of the equations. A solution to a system of equations must satisfy both equations in the system.

Q: How do I know if an equation holds true?

A: An equation holds true if the left-hand side of the equation is equal to the right-hand side of the equation.

Q: What if I get a false statement when I substitute the values of the ordered pair into an equation?

A: If you get a false statement when you substitute the values of the ordered pair into an equation, then the ordered pair is not a solution to the system of equations.

Q: Can I use a calculator to verify a solution to a system of equations?

A: Yes, you can use a calculator to verify a solution to a system of equations. However, you should always check your work by hand to make sure that the calculator is giving you the correct answer.

Q: What if I am given a system of equations with variables and constants on both sides?

A: If you are given a system of equations with variables and constants on both sides, you can still use the same process to verify a solution to the system. Simply substitute the values of the ordered pair into each equation and check if the equation holds true.

Q: Can I use the same process to verify a solution to a system of equations with more than two equations?

A: Yes, you can use the same process to verify a solution to a system of equations with more than two equations. Simply substitute the values of the ordered pair into each equation and check if the equation holds true.

Conclusion

In this article, we answered some frequently asked questions about verifying solutions to a system of equations. We discussed the process of verifying a solution to a system of equations, including substituting the values of the ordered pair into each equation and checking if the equation holds true. We also provided examples and solutions to help illustrate the process. By following these steps, you can verify if a given ordered pair is a solution to a system of equations.

Example Problems

Here are a few example problems that involve verifying solutions to a system of equations:

1. Verify if the ordered pair $\left(2,3\right)$ is a solution to the system of equations:

$$\begin{align*} y &= 2x - 1 \\ y &= x + 2 \end{align*}$$

2. Verify if the ordered pair $\left(-1,2\right)$ is a solution to the system of equations:

$$\begin{align*} y &= -x + 3 \\ y &= 2x - 1 \end{align*}$$

3. Verify if the ordered pair $\left(0,0\right)$ is a solution to the system of equations:

$$\begin{align*} y &= 2x - 1 \\ y &= -x + 1 \end{align*}$$

Solutions

1. To verify if the ordered pair $\left(2,3\right)$ is a solution to the system of equations, we need to substitute the values of $x$ and $y$ into each equation and check if the equation holds true.

Substituting into the first equation:

$$y = 2x - 1$$

$$3 = 2(2) - 1$$

$$3 = 3$$

The equation holds true, which means that the ordered pair $\left(2,3\right)$ satisfies the first equation.

Substituting into the second equation:

$$y = x + 2$$

$$3 = 2 + 2$$

$$3 = 4$$

The equation does not hold true, which means that the ordered pair $\left(2,3\right)$ does not satisfy the second equation.

Therefore, the ordered pair $\left(2,3\right)$ is not a solution to the system of equations.

2. To verify if the ordered pair $\left(-1,2\right)$ is a solution to the system of equations, we need to substitute the values of $x$ and $y$ into each equation and check if the equation holds true.

Substituting into the first equation:

$$y = -x + 3$$

$$2 = -(-1) + 3$$

$$2 = 4$$

The equation does not hold true, which means that the ordered pair $\left(-1,2\right)$ does not satisfy the first equation.

Substituting into the second equation:

$$y = 2x - 1$$

$$2 = 2(-1) - 1$$

$$2 = -3$$

The equation does not hold true, which means that the ordered pair $\left(-1,2\right)$ does not satisfy the second equation.

Therefore, the ordered pair $\left(-1,2\right)$ is not a solution to the system of equations.

3. To verify if the ordered pair $\left(0,0\right)$ is a solution to the system of equations, we need to substitute the values of $x$ and $y$ into each equation and check if the equation holds true.

Substituting into the first equation:

$$y = 2x - 1$$

$$0 = 2(0) - 1$$

$$0 = -1$$

The equation does not hold true, which means that the ordered pair $\left(0,0\right)$ does not satisfy the first equation.

Substituting into the second equation:

$$y = -x + 1$$

$$0 = -0 + 1$$

$$0 = 1$$

The equation does not hold true, which means that the ordered pair $\left(0,0\right)$ does not satisfy the second equation.

Therefore, the ordered pair $\left(0,0\right)$ is not a solution to the system of equations.

Conclusion

In this article, we answered some frequently asked questions about verifying solutions to a system of equations. We discussed the process of verifying a solution to a system of equations, including substituting the values of the ordered pair into each equation and checking if the equation holds true. We also provided examples and solutions to help illustrate the process. By following these steps, you can verify if a given ordered pair is a solution to a system of equations.