Solve The System By Substitution.${ \begin{align*} y &= -6x - 7 \ y &= -7x \end{align*} }$

Introduction

In this article, we will explore the method of substitution to solve a system of linear equations. This method involves solving one equation for a variable and then substituting that expression into the other equation. We will use a system of two linear equations in two variables to demonstrate this method.

The System of Linear Equations

The system of linear equations we will be working with is:

${ y = -6x - 7 }$ ${ y = -7x }$

Step 1: Solve One Equation for a Variable

To solve this system by substitution, we need to solve one equation for a variable. Let's solve the second equation for y:

${ y = -7x }$

We can rewrite this equation as:

${ -7x = y }$

Now, we can solve for x by dividing both sides of the equation by -7:

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

Step 2: Substitute the Expression into the Other Equation

Now that we have solved the second equation for x, we can substitute this expression into the first equation. The first equation is:

${ y = -6x - 7 }$

We can substitute the expression for x that we found in Step 1 into this equation:

${ y = -6\left(-\frac{y}{7}\right) - 7 }$

Step 3: Simplify the Equation

Now, we can simplify the equation by multiplying the terms inside the parentheses:

${ y = \frac{6y}{7} - 7 }$

Next, we can multiply both sides of the equation by 7 to eliminate the fraction:

${ 7y = 6y - 49 }$

Step 4: Solve for the Variable

Now, we can solve for y by subtracting 6y from both sides of the equation:

${ 7y - 6y = -49 }$

This simplifies to:

${ y = -49 }$

Step 5: Find the Value of the Other Variable

Now that we have found the value of y, we can substitute this value into one of the original equations to find the value of the other variable. Let's use the second equation:

${ y = -7x }$

We can substitute y = -49 into this equation:

${ -49 = -7x }$

Now, we can solve for x by dividing both sides of the equation by -7:

${ x = \frac{-49}{-7} }$

This simplifies to:

${ x = 7 }$

Conclusion

In this article, we used the method of substitution to solve a system of linear equations. We solved one equation for a variable and then substituted that expression into the other equation. We then simplified the resulting equation and solved for the variable. Finally, we found the value of the other variable by substituting the value of the first variable into one of the original equations.

Example Problems

Here are a few example problems that demonstrate the method of substitution:

Example 1

Solve the system of linear equations:

${ y = 2x + 3 }$ ${ y = 3x - 2 }$

Solution

To solve this system by substitution, we need to solve one equation for a variable. Let's solve the second equation for y:

${ y = 3x - 2 }$

We can rewrite this equation as:

${ 3x - 2 = y }$

Now, we can solve for x by adding 2 to both sides of the equation and then dividing both sides by 3:

${ 3x = y + 2 }$ ${ x = \frac{y + 2}{3} }$

Now, we can substitute the expression for x into the first equation:

${ y = 2\left(\frac{y + 2}{3}\right) + 3 }$

We can simplify this equation by multiplying the terms inside the parentheses:

${ y = \frac{2y + 4}{3} + 3 }$

Next, we can multiply both sides of the equation by 3 to eliminate the fraction:

${ 3y = 2y + 4 + 9 }$

This simplifies to:

${ 3y = 2y + 13 }$

Now, we can solve for y by subtracting 2y from both sides of the equation:

${ 3y - 2y = 13 }$

This simplifies to:

${ y = 13 }$

Now that we have found the value of y, we can substitute this value into one of the original equations to find the value of the other variable. Let's use the second equation:

${ y = 3x - 2 }$

We can substitute y = 13 into this equation:

${ 13 = 3x - 2 }$

Now, we can solve for x by adding 2 to both sides of the equation and then dividing both sides by 3:

${ 3x = 13 + 2 }$ ${ 3x = 15 }$ ${ x = \frac{15}{3} }$

This simplifies to:

${ x = 5 }$

Example 2

Solve the system of linear equations:

${ y = -4x + 2 }$ ${ y = -2x - 3 }$

Solution

To solve this system by substitution, we need to solve one equation for a variable. Let's solve the second equation for y:

${ y = -2x - 3 }$

We can rewrite this equation as:

${ -2x - 3 = y }$

Now, we can solve for x by adding 3 to both sides of the equation and then dividing both sides by -2:

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

Now, we can substitute the expression for x into the first equation:

${ y = -4\left(-\frac{y + 3}{2}\right) + 2 }$

We can simplify this equation by multiplying the terms inside the parentheses:

${ y = \frac{4y + 12}{2} + 2 }$

Next, we can multiply both sides of the equation by 2 to eliminate the fraction:

${ 2y = 4y + 12 + 4 }$

This simplifies to:

${ 2y = 4y + 16 }$

Now, we can solve for y by subtracting 4y from both sides of the equation:

${ 2y - 4y = 16 }$

This simplifies to:

${ -2y = 16 }$

Now, we can solve for y by dividing both sides of the equation by -2:

${ y = -\frac{16}{2} }$

This simplifies to:

${ y = -8 }$

Now that we have found the value of y, we can substitute this value into one of the original equations to find the value of the other variable. Let's use the second equation:

${ y = -2x - 3 }$

We can substitute y = -8 into this equation:

${ -8 = -2x - 3 }$

Now, we can solve for x by adding 3 to both sides of the equation and then dividing both sides by -2:

${ -2x = -8 + 3 }$ ${ -2x = -5 }$ ${ x = \frac{-5}{-2} }$

This simplifies to:

${ x = \frac{5}{2} }$

Example 3

Solve the system of linear equations:

${ y = 3x - 2 }$ ${ y = 2x + 1 }$

Solution

To solve this system by substitution, we need to solve one equation for a variable. Let's solve the second equation for y:

${ y = 2x + 1 }$

We can rewrite this equation as:

${ 2x + 1 = y }$

Now, we can solve for x by subtracting 1 from both sides of the equation and then dividing both sides by 2:

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

Now, we can substitute the expression for x into the first equation:

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

We can simplify this equation by multiplying the terms inside the parentheses:

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

Next, we can multiply both sides of the equation by 2 to eliminate the fraction:

${ 2y = 3y - 3 - 4 }$

This simplifies to:

${ 2y = 3y - 7 }$

Now, we can solve for y by subtracting 3y from both sides of the equation:

${ 2y - 3y = -7 }$

This simplifies to:

${ -y = -7 }$

Now, we can solve for y by dividing both sides of the equation by -1:

${ y = 7 }$

Now that we have found the value of y, we can substitute this value into one of the original equations to find the value of the other variable. Let's use the second equation:

${ y = 2x + 1 }$

We can substitute y = 7 into this equation:

${ 7 = 2x + 1 }$

Introduction

In our previous article, we explored the method of substitution to solve a system of linear equations. We solved one equation for a variable and then substituted that expression into the other equation. We then simplified the resulting equation and solved for the variable. In this article, we will answer some common questions about solving a system of linear equations by substitution.

Q: What is the method of substitution?

A: The method of substitution is a technique used to solve a system of linear equations. It involves solving one equation for a variable and then substituting that expression into the other equation.

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

A: To determine which equation to solve for a variable, you can look at the coefficients of the variables in both equations. If one equation has a coefficient of 1 for one variable, it is often easier to solve for that variable.

Q: What if I get a fraction when I substitute the expression into the other equation?

A: If you get a fraction when you substitute the expression into the other equation, you can multiply both sides of the equation by the denominator of the fraction to eliminate the fraction.

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

A: Yes, you can use the method of substitution to solve a system of linear equations with three variables. However, it may be more complicated and require more steps.

Q: What if I get a quadratic equation when I solve for the variable?

A: If you get a quadratic equation when you solve for the variable, you can use the quadratic formula to solve for the variable.

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

A: Yes, you can use the method of substitution to solve a system of linear equations with decimals. However, you may need to use a calculator to perform the calculations.

Q: What if I get a negative value for the variable?

A: If you get a negative value for the variable, it is still a valid solution to the system of linear equations.

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

A: Yes, you can use the method of substitution to solve a system of linear equations with fractions. However, you may need to use a calculator to perform the calculations.

Q: What if I get a complex number as a solution?

A: If you get a complex number as a solution, it is still a valid solution to the system of linear equations.

Example Problems

Here are a few example problems that demonstrate the method of substitution:

Example 1

Solve the system of linear equations:

${ y = 2x + 3 }$ ${ y = 3x - 2 }$

Solution

To solve this system by substitution, we need to solve one equation for a variable. Let's solve the second equation for y:

${ y = 3x - 2 }$

We can rewrite this equation as:

${ 3x - 2 = y }$

Now, we can solve for x by adding 2 to both sides of the equation and then dividing both sides by 3:

${ 3x = y + 2 }$ ${ x = \frac{y + 2}{3} }$

Now, we can substitute the expression for x into the first equation:

${ y = 2\left(\frac{y + 2}{3}\right) + 3 }$

We can simplify this equation by multiplying the terms inside the parentheses:

${ y = \frac{2y + 4}{3} + 3 }$

Next, we can multiply both sides of the equation by 3 to eliminate the fraction:

${ 3y = 2y + 4 + 9 }$

This simplifies to:

${ 3y = 2y + 13 }$

Now, we can solve for y by subtracting 2y from both sides of the equation:

${ 3y - 2y = 13 }$

This simplifies to:

${ y = 13 }$

Now that we have found the value of y, we can substitute this value into one of the original equations to find the value of the other variable. Let's use the second equation:

${ y = 3x - 2 }$

We can substitute y = 13 into this equation:

${ 13 = 3x - 2 }$

Now, we can solve for x by adding 2 to both sides of the equation and then dividing both sides by 3:

${ 3x = 13 + 2 }$ ${ 3x = 15 }$ ${ x = \frac{15}{3} }$

This simplifies to:

${ x = 5 }$

Example 2

Solve the system of linear equations:

${ y = -4x + 2 }$ ${ y = -2x - 3 }$

Solution

To solve this system by substitution, we need to solve one equation for a variable. Let's solve the second equation for y:

${ y = -2x - 3 }$

We can rewrite this equation as:

${ -2x - 3 = y }$

Now, we can solve for x by adding 3 to both sides of the equation and then dividing both sides by -2:

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

Now, we can substitute the expression for x into the first equation:

${ y = -4\left(-\frac{y + 3}{2}\right) + 2 }$

We can simplify this equation by multiplying the terms inside the parentheses:

${ y = \frac{4y + 12}{2} + 2 }$

Next, we can multiply both sides of the equation by 2 to eliminate the fraction:

${ 2y = 4y + 12 + 4 }$

This simplifies to:

${ 2y = 4y + 16 }$

Now, we can solve for y by subtracting 4y from both sides of the equation:

${ 2y - 4y = 16 }$

This simplifies to:

${ -2y = 16 }$

Now, we can solve for y by dividing both sides of the equation by -2:

${ y = -\frac{16}{2} }$

This simplifies to:

${ y = -8 }$

Now that we have found the value of y, we can substitute this value into one of the original equations to find the value of the other variable. Let's use the second equation:

${ y = -2x - 3 }$

We can substitute y = -8 into this equation:

${ -8 = -2x - 3 }$

Now, we can solve for x by adding 3 to both sides of the equation and then dividing both sides by -2:

${ -2x = -8 + 3 }$ ${ -2x = -5 }$ ${ x = \frac{-5}{-2} }$

This simplifies to:

${ x = \frac{5}{2} }$

Example 3

Solve the system of linear equations:

${ y = 3x - 2 }$ ${ y = 2x + 1 }$

Solution

To solve this system by substitution, we need to solve one equation for a variable. Let's solve the second equation for y:

${ y = 2x + 1 }$

We can rewrite this equation as:

${ 2x + 1 = y }$

Now, we can solve for x by subtracting 1 from both sides of the equation and then dividing both sides by 2:

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

Now, we can substitute the expression for x into the first equation:

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

We can simplify this equation by multiplying the terms inside the parentheses:

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

Next, we can multiply both sides of the equation by 2 to eliminate the fraction:

${ 2y = 3y - 3 - 4 }$

This simplifies to:

${ 2y = 3y - 7 }$

Now, we can solve for y by subtracting 3y from both sides of the equation:

${ 2y - 3y = -7 }$

This simplifies to:

${ -y = -7 }$

Now, we can solve for y by dividing both sides of the equation by -1:

${ y = 7 }$

Now that we have found the value of y, we can substitute this value into one of the original equations to find the value of the other variable. Let's use the second equation:

${ y = 2x + 1 }$

We can substitute y = 7 into this equation:

${ 7 = 2x + 1 }$

Now, we can solve for x by subtracting 1 from both sides of the equation and then dividing both sides by 2:

${ 2x = 7 - 1 }$ ${ 2x = 6 }$ ${ x = \frac{6}{2} }$

This simplifies to:

${ x = 3 }$

Conclusion

In this article, we answered some common questions about