Jerry Solved The System Of Equations:$\[ \begin{array}{r} x - 3y = 1 \\ 7x + 2y = 7 \end{array} \\]As The First Step, He Decided To Solve For \[$ Y \$\] In The Second Equation Because It Had The Smallest Number As A Coefficient. Max

Introduction

Solving systems of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will explore how to solve a system of linear equations using the method of substitution and elimination. We will use the example of Jerry, who is trying to solve the system of equations:

$$\begin{array}{r} x - 3y = 1 \\ 7x + 2y = 7 \end{array}$$

Step 1: Choose the Equation with the Smallest Coefficient

The first step in solving a system of equations is to choose the equation with the smallest coefficient. In this case, Jerry decided to solve for $y$ in the second equation because it had the smallest number as a coefficient. The second equation is:

$$7x + 2y = 7$$

Solving for $y$

To solve for $y$, we need to isolate the variable $y$ on one side of the equation. We can do this by subtracting $7x$ from both sides of the equation:

$$2y = 7 - 7x$$

Next, we can divide both sides of the equation by 2 to solve for $y$:

$$y = \frac{7 - 7x}{2}$$

Substituting the Value of $y$ into the First Equation

Now that we have the value of $y$, we can substitute it into the first equation to solve for $x$. The first equation is:

$$x - 3y = 1$$

We can substitute the value of $y$ into this equation:

$$x - 3\left(\frac{7 - 7x}{2}\right) = 1$$

Simplifying the Equation

To simplify the equation, we can start by multiplying both sides of the equation by 2 to eliminate the fraction:

$$2x - 3(7 - 7x) = 2$$

Next, we can distribute the 3 to the terms inside the parentheses:

$$2x - 21 + 21x = 2$$

Combining Like Terms

We can combine like terms by adding $2x$ and $21x$:

$$23x - 21 = 2$$

Adding 21 to Both Sides

To isolate the term with $x$, we can add 21 to both sides of the equation:

$$23x = 2 + 21$$

$$23x = 23$$

Dividing Both Sides by 23

Finally, we can divide both sides of the equation by 23 to solve for $x$:

$$x = \frac{23}{23}$$

$$x = 1$$

Finding the Value of $y$

Now that we have the value of $x$, we can substitute it into the equation for $y$ to find the value of $y$:

$$y = \frac{7 - 7x}{2}$$

$$y = \frac{7 - 7(1)}{2}$$

$$y = \frac{7 - 7}{2}$$

$$y = \frac{0}{2}$$

$$y = 0$$

Conclusion

In this article, we have shown how to solve a system of linear equations using the method of substitution and elimination. We have used the example of Jerry, who is trying to solve the system of equations:

$$\begin{array}{r} x - 3y = 1 \\ 7x + 2y = 7 \end{array}$$

We have shown how to choose the equation with the smallest coefficient, solve for $y$, substitute the value of $y$ into the first equation, simplify the equation, combine like terms, add 21 to both sides, divide both sides by 23, and find the value of $y$. We have found that the values of $x$ and $y$ are $x = 1$ and $y = 0$.

Real-World Applications

Solving systems of equations has many real-world applications, including:

• Physics: Solving systems of equations is used to describe the motion of objects in physics.
• Engineering: Solving systems of equations is used to design and optimize systems in engineering.
• Economics: Solving systems of equations is used to model economic systems and make predictions about economic trends.
• Computer Science: Solving systems of equations is used in computer science to solve problems in computer graphics, game development, and machine learning.

Tips and Tricks

Here are some tips and tricks for solving systems of equations:

• Choose the equation with the smallest coefficient: This will make it easier to solve for the variable.
• Use the method of substitution and elimination: This will make it easier to solve for the variables.
• Simplify the equation: This will make it easier to solve for the variables.
• Combine like terms: This will make it easier to solve for the variables.
• Add or subtract the same value to both sides: This will make it easier to solve for the variables.

Conclusion

Solving systems of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we have shown how to solve a system of linear equations using the method of substitution and elimination. We have used the example of Jerry, who is trying to solve the system of equations:

$$\begin{array}{r} x - 3y = 1 \\ 7x + 2y = 7 \end{array}$$

$$$$$$$$$$$$$$

Introduction

Solving systems of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In our previous article, we showed how to solve a system of linear equations using the method of substitution and elimination. In this article, we will answer some frequently asked questions about solving systems of equations.

Q: What is a system of equations?

A system of equations is a set of two or more equations that involve the same variables. For example:

$$\begin{array}{r} x - 3y = 1 \\ 7x + 2y = 7 \end{array}$$

Q: How do I choose the equation with the smallest coefficient?

To choose the equation with the smallest coefficient, you need to look at the coefficients of the variables in each equation. The coefficient is the number that is multiplied by the variable. For example, in the equation $x - 3y = 1$, the coefficient of $x$ is 1 and the coefficient of $y$ is -3. In the equation $7x + 2y = 7$, the coefficient of $x$ is 7 and the coefficient of $y$ is 2. You should choose the equation with the smallest coefficient.

Q: How do I solve for $y$?

To solve for $y$, you need to isolate the variable $y$ on one side of the equation. You can do this by subtracting the product of the coefficient of $x$ and the variable $x$ from both sides of the equation. For example, in the equation $7x + 2y = 7$, you can subtract $7x$ from both sides to get:

$$2y = 7 - 7x$$

Next, you can divide both sides of the equation by 2 to solve for $y$:

$$y = \frac{7 - 7x}{2}$$

Q: How do I substitute the value of $y$ into the first equation?

To substitute the value of $y$ into the first equation, you need to replace the variable $y$ with the expression you found in the previous step. For example, if you found that $y = \frac{7 - 7x}{2}$, you can substitute this expression into the first equation:

$$x - 3\left(\frac{7 - 7x}{2}\right) = 1$$

Q: How do I simplify the equation?

To simplify the equation, you need to combine like terms and eliminate any fractions. For example, in the equation $x - 3\left(\frac{7 - 7x}{2}\right) = 1$, you can multiply both sides of the equation by 2 to eliminate the fraction:

$$2x - 3(7 - 7x) = 2$$

Next, you can distribute the 3 to the terms inside the parentheses:

$$2x - 21 + 21x = 2$$

Q: How do I combine like terms?

To combine like terms, you need to add or subtract the same value to both sides of the equation. For example, in the equation $2x - 21 + 21x = 2$, you can combine the like terms $2x$ and $21x$:

$$23x - 21 = 2$$

Q: How do I add or subtract the same value to both sides?

To add or subtract the same value to both sides, you need to add or subtract the same value to both sides of the equation. For example, in the equation $23x - 21 = 2$, you can add 21 to both sides to get:

$$23x = 2 + 21$$

$$23x = 23$$

Q: How do I divide both sides by 23?

To divide both sides by 23, you need to divide both sides of the equation by 23. For example, in the equation $23x = 23$, you can divide both sides by 23 to solve for $x$:

$$x = \frac{23}{23}$$

$$x = 1$$

Q: How do I find the value of $y$?

To find the value of $y$, you need to substitute the value of $x$ into the equation for $y$. For example, if you found that $y = \frac{7 - 7x}{2}$, you can substitute $x = 1$ into this equation:

$$y = \frac{7 - 7(1)}{2}$$

$$y = \frac{7 - 7}{2}$$

$$y = \frac{0}{2}$$

$$y = 0$$

Conclusion

Solving systems of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we have answered some frequently asked questions about solving systems of equations. We have shown how to choose the equation with the smallest coefficient, solve for $y$, substitute the value of $y$ into the first equation, simplify the equation, combine like terms, add or subtract the same value to both sides, divide both sides by 23, and find the value of $y$. We have found that the values of $x$ and $y$ are $x = 1$ and $y = 0$.

Real-World Applications

Solving systems of equations has many real-world applications, including:

• Physics: Solving systems of equations is used to describe the motion of objects in physics.
• Engineering: Solving systems of equations is used to design and optimize systems in engineering.
• Economics: Solving systems of equations is used to model economic systems and make predictions about economic trends.
• Computer Science: Solving systems of equations is used in computer science to solve problems in computer graphics, game development, and machine learning.

Tips and Tricks

Here are some tips and tricks for solving systems of equations:

• Choose the equation with the smallest coefficient: This will make it easier to solve for the variable.
• Use the method of substitution and elimination: This will make it easier to solve for the variables.
• Simplify the equation: This will make it easier to solve for the variables.
• Combine like terms: This will make it easier to solve for the variables.
• Add or subtract the same value to both sides: This will make it easier to solve for the variables.
• Divide both sides by the coefficient: This will make it easier to solve for the variable.

Conclusion

Solving systems of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we have answered some frequently asked questions about solving systems of equations. We have shown how to choose the equation with the smallest coefficient, solve for $y$, substitute the value of $y$ into the first equation, simplify the equation, combine like terms, add or subtract the same value to both sides, divide both sides by 23, and find the value of $y$. We have found that the values of $x$ and $y$ are $x = 1$ and $y = 0$. Solving systems of equations has many real-world applications, including physics, engineering, economics, and computer science. We have also provided some tips and tricks for solving systems of equations.