Solve The System Of Equations.$\[ \begin{align*} y &= 2x - 1 \\ y &= 5 - X \end{align*} \\]Graph The Equation \[$y = 5 - X\$\].

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. In this article, we will focus on solving a system of two linear equations in two variables. We will use the method of substitution to solve the system of equations and then graph the equation $y = 5 - x$.

The System of Equations

The system of equations we will be solving is:

$$\begin{align*} y &= 2x - 1 \\ y &= 5 - x \end{align*}$$

Step 1: Write Down the Equations

The first step in solving the system of equations is to write down the equations. In this case, we have two equations:

$$\begin{align*} y &= 2x - 1 \\ y &= 5 - x \end{align*}$$

Step 2: Set the Equations Equal to Each Other

Since both equations are equal to $y$, we can set them equal to each other:

$$2x - 1 = 5 - x$$

Step 3: Add $x$ to Both Sides

To get rid of the negative term, we can add $x$ to both sides of the equation:

$$3x - 1 = 5$$

Step 4: Add 1 to Both Sides

To isolate the term with the variable, we can add 1 to both sides of the equation:

$$3x = 6$$

Step 5: Divide Both Sides by 3

To solve for $x$, we can divide both sides of the equation by 3:

$$x = 2$$

Step 6: Substitute $x$ into One of the Original Equations

Now that we have found the value of $x$, we can substitute it into one of the original equations to find the value of $y$. We will use the first equation:

$$y = 2x - 1$$

Substituting $x = 2$ into the equation, we get:

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

$$y = 4 - 1$$

$$y = 3$$

The Solution

The solution to the system of equations is $x = 2$ and $y = 3$.

Graphing the Equation $y = 5 - x$

To graph the equation $y = 5 - x$, we can use the slope-intercept form of a linear equation, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

In this case, the equation is $y = 5 - x$, which can be rewritten as $y = -x + 5$. The slope of the equation is $-1$, and the y-intercept is $5$.

To graph the equation, we can start by plotting the y-intercept, which is the point $(0, 5)$. Then, we can use the slope to find another point on the line. Since the slope is $-1$, we can move down 1 unit and to the right 1 unit from the y-intercept to find the next point.

The next point on the line is $(1, 4)$. We can continue this process to find more points on the line.

Conclusion

In this article, we solved a system of two linear equations in two variables using the method of substitution. We then graphed the equation $y = 5 - x$ using the slope-intercept form of a linear equation. The solution to the system of equations is $x = 2$ and $y = 3$.

Key Takeaways

• A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.
• The method of substitution is a technique used to solve systems of equations.
• The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• To graph a linear equation, we can start by plotting the y-intercept and then use the slope to find other points on the line.

Further Reading

If you want to learn more about solving systems of equations and graphing linear equations, I recommend checking out the following resources:

• Khan Academy: Solving Systems of Equations
• Mathway: Solving Systems of Equations
• Purplemath: Graphing Linear Equations

References

• Larson, R., & Hostetler, R. P. (2015). College Algebra. Cengage Learning.
• Sullivan, M. (2015). College Algebra. Pearson Education.

Glossary

• System of equations: A set of two or more equations that are solved simultaneously to find the values of the variables.
• Method of substitution: A technique used to solve systems of equations.
• Slope-intercept form: A form of a linear equation that is written as $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• Graphing: The process of visualizing a mathematical equation or function using a coordinate plane.
  Solving the System of Equations: A Q&A Guide =====================================================

Introduction

In our previous article, we solved a system of two linear equations in two variables using the method of substitution. We also graphed the equation $y = 5 - x$ using the slope-intercept form of a linear equation. In this article, we will answer some frequently asked questions about solving systems of equations and graphing linear equations.

Q&A

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.

Q: What is the method of substitution?

A: The method of substitution is a technique used to solve systems of equations. It involves substituting one equation into the other equation to eliminate one of the variables.

Q: How do I know which method to use to solve a system of equations?

A: There are two main methods used to solve systems of equations: substitution and elimination. The method you choose will depend on the equations you are working with. If one equation is already solved for one of the variables, you can use the substitution method. If not, you can use the elimination method.

Q: What is the slope-intercept form of a linear equation?

A: The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: How do I graph a linear equation?

A: To graph a linear equation, you can start by plotting the y-intercept and then use the slope to find other points on the line. You can also use a graphing calculator or software to graph the equation.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. A quadratic equation is an equation in which the highest power of the variable is 2.

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

A: No, the method of substitution is only used to solve systems of linear equations. To solve a system of quadratic equations, you will need to use a different method, such as factoring or the quadratic formula.

Q: How do I know if a system of equations has a solution?

A: If a system of equations has a solution, it means that there is a set of values for the variables that satisfies both equations. If a system of equations does not have a solution, it means that there is no set of values for the variables that satisfies both equations.

Q: What is the difference between a system of equations and a system of inequalities?

A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. A system of inequalities is a set of two or more inequalities that are solved simultaneously to find the values of the variables.

Q: Can I use a graphing calculator or software to solve a system of equations?

A: Yes, you can use a graphing calculator or software to solve a system of equations. Many graphing calculators and software programs have built-in functions for solving systems of equations.

Conclusion

In this article, we answered some frequently asked questions about solving systems of equations and graphing linear equations. We hope that this article has been helpful in clarifying any confusion you may have had about these topics.

Key Takeaways

• A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.
• The method of substitution is a technique used to solve systems of equations.
• The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• To graph a linear equation, you can start by plotting the y-intercept and then use the slope to find other points on the line.
• A linear equation is an equation in which the highest power of the variable is 1.
• A quadratic equation is an equation in which the highest power of the variable is 2.

Further Reading

If you want to learn more about solving systems of equations and graphing linear equations, I recommend checking out the following resources:

• Khan Academy: Solving Systems of Equations
• Mathway: Solving Systems of Equations
• Purplemath: Graphing Linear Equations

References

• Larson, R., & Hostetler, R. P. (2015). College Algebra. Cengage Learning.
• Sullivan, M. (2015). College Algebra. Pearson Education.

Glossary

• System of equations: A set of two or more equations that are solved simultaneously to find the values of the variables.
• Method of substitution: A technique used to solve systems of equations.
• Slope-intercept form: A form of a linear equation that is written as $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• Graphing: The process of visualizing a mathematical equation or function using a coordinate plane.