Considering These Two Equations:$\[ \begin{array}{l} y - 4x = 6 \\ 8y + 5x = 6 \end{array} \\]Which Expression Can Be Substituted For \[$ Y \$\] In The Bottom Equation Of The System To Solve The System By Substitution?A. \[$ 4x + 6

Introduction

In mathematics, a system of linear equations is a set of two or more equations that involve two or more variables. Solving a system of linear equations is an essential skill in algebra and is used to find the values of the variables that satisfy all the equations in the system. There are several methods to solve a system of linear equations, including substitution, elimination, and graphing. In this article, we will focus on solving a system of linear equations by substitution.

What is Substitution Method?

The substitution method is a technique used to solve a system of linear equations by substituting the expression for one variable from one equation into the other equation. This method is useful when one of the equations is already solved for one variable. The substitution method involves the following steps:

1. Solve one of the equations for one variable.
2. Substitute the expression for the variable into the other equation.
3. Solve the resulting equation for the other variable.
4. Back-substitute the value of the variable into one of the original equations to find the value of the other variable.

Given Equations

The given system of linear equations is:

$$\begin{array}{l} y - 4x = 6 \\ 8y + 5x = 6 \end{array}$$

Step 1: Solve One of the Equations for One Variable

To solve the system by substitution, we need to solve one of the equations for one variable. Let's solve the first equation for y:

$$y - 4x = 6$$

$$y = 4x + 6$$

Now, we have an expression for y in terms of x.

Step 2: Substitute the Expression for y into the Other Equation

Substitute the expression for y into the second equation:

$$8y + 5x = 6$$

$$8(4x + 6) + 5x = 6$$

Expand and simplify the equation:

$$32x + 48 + 5x = 6$$

Combine like terms:

$$37x + 48 = 6$$

Subtract 48 from both sides:

$$37x = -42$$

Divide both sides by 37:

$$x = -\frac{42}{37}$$

Step 3: Back-Substitute the Value of x into One of the Original Equations

Now that we have the value of x, we can back-substitute it into one of the original equations to find the value of y. Let's use the first equation:

$$y - 4x = 6$$

$$y - 4\left(-\frac{42}{37}\right) = 6$$

Simplify the equation:

$$y + \frac{168}{37} = 6$$

Subtract $\frac{168}{37}$ from both sides:

$$y = 6 - \frac{168}{37}$$

Simplify the fraction:

$$y = \frac{222 - 168}{37}$$

$$y = \frac{54}{37}$$

Conclusion

In this article, we solved a system of linear equations by substitution. We first solved one of the equations for one variable, then substituted the expression for the variable into the other equation. We solved the resulting equation for the other variable and back-substituted the value of the variable into one of the original equations to find the value of the other variable. The substitution method is a useful technique for solving systems of linear equations, especially when one of the equations is already solved for one variable.

Answer

The expression that can be substituted for y in the bottom equation of the system to solve the system by substitution is:

$$4x + 6$$

Q: What is the substitution method for solving systems of linear equations?

A: The substitution method is a technique used to solve a system of linear equations by substituting the expression for one variable from one equation into the other equation. This method is useful when one of the equations is already solved for one variable.

Q: How do I know which equation to solve for one variable first?

A: You can choose either equation to solve for one variable first. However, it's often easier to solve the equation that has the variable with the smallest coefficient first.

Q: What if I have a system of linear equations with three or more variables? Can I still use the substitution method?

A: Yes, you can still use the substitution method to solve a system of linear equations with three or more variables. However, you will need to substitute the expression for one variable into the other equations multiple times, until you have solved for all the variables.

Q: What if I get stuck or make a mistake while solving a system of linear equations by substitution?

A: Don't worry! It's easy to get stuck or make a mistake while solving a system of linear equations by substitution. If you get stuck, try going back to the previous step and re-checking your work. If you make a mistake, try to identify where you went wrong and correct it.

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

A: Yes, you can use the substitution method to solve a system of linear equations with fractions or decimals. However, you will need to be careful when multiplying and dividing fractions or decimals.

Q: Is the substitution method the only way to solve a system of linear equations?

A: No, there are several other methods for solving a system of linear equations, including the elimination method and the graphing method. The substitution method is just one of the many techniques you can use to solve a system of linear equations.

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

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

Q: How do I know if I have solved a system of linear equations correctly?

A: To check if you have solved a system of linear equations correctly, plug the values of the variables back into the original equations and make sure they are true. If the equations are true, then you have solved the system correctly.

Q: Can I use a calculator to solve a system of linear equations by substitution?

A: Yes, you can use a calculator to solve a system of linear equations by substitution. However, be careful when entering the equations and making sure the calculator is set to the correct mode.

Q: Is the substitution method a good way to solve a system of linear equations with many variables?

A: The substitution method can be a good way to solve a system of linear equations with many variables, but it can also be time-consuming and prone to errors. In some cases, it may be better to use a different method, such as the elimination method or a numerical method.

Conclusion

In this article, we answered some frequently asked questions about solving systems of linear equations by substitution. We covered topics such as the substitution method, choosing which equation to solve for one variable first, and using the substitution method with fractions or decimals. We also discussed the limitations of the substitution method and when it may be better to use a different method.