Substituting x into the Second Equation

Oct 13, 2025 by talynt.org 148 views

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Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will consider a system of two linear equations with two variables and determine the number of solutions it has.

The System of Equations

The system of equations we will consider is:

${ \begin{align*} 8x + 2y &= 8 \\ y &= -4x + 4 \end{align*} }$

This system consists of two linear equations with two variables, x and y. The first equation is:

${ 8x + 2y = 8 }$

And the second equation is:

${ y = -4x + 4 }$

Substitution Method

One way to solve this system is by using the substitution method. This method involves solving one of the equations for one variable and then substituting that expression into the other equation. In this case, we can solve the second equation for y:

${ y = -4x + 4 }$

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

${ 8x + 2(-4x + 4) = 8 }$

Simplifying the Equation

To simplify the equation, we can start by distributing the 2 to the terms inside the parentheses:

${ 8x - 8x + 8 = 8 }$

This simplifies to:

${ 8 = 8 }$

The Result

As we can see, the equation is an identity, which means that it is true for all values of x. This means that the system of equations has infinitely many solutions.

Geometric Interpretation

Geometrically, the system of equations can be represented as a pair of lines on a coordinate plane. The first equation represents a line with a slope of -4 and a y-intercept of 4. The second equation represents a line with a slope of -4 and a y-intercept of 4 as well. Since the two lines are identical, they intersect at every point on the line.

Conclusion

In conclusion, the system of equations has infinitely many solutions. This is because the two equations represent the same line, and therefore, every point on the line is a solution to the system.

Example 2

Consider the system of equations:

${ \begin{align*} x + 2y &= 6 \\ y &= -\frac{1}{2}x + 3 \end{align*} }$

This system consists of two linear equations with two variables, x and y. The first equation is:

${ x + 2y = 6 }$

And the second equation is:

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

Substitution Method

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

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

${ x + 2(-\frac{1}{2}x + 3) = 6 }$

Simplifying the Equation

To simplify the equation, we can start by distributing the 2 to the terms inside the parentheses:

${ x - x + 6 = 6 }$

This simplifies to:

${ 6 = 6 }$

The Result

As we can see, the equation is an identity, which means that it is true for all values of x. This means that the system of equations has infinitely many solutions.

Geometric Interpretation

Geometrically, the system of equations can be represented as a pair of lines on a coordinate plane. The first equation represents a line with a slope of 2 and a y-intercept of 3. The second equation represents a line with a slope of -1/2 and a y-intercept of 3. Since the two lines are parallel, they do not intersect, and therefore, the system of equations has no solution.

Conclusion

In conclusion, the system of equations has no solution. This is because the two equations represent parallel lines, and therefore, they do not intersect.

Example 3

Consider the system of equations:

${ \begin{align*} x + 2y &= 6 \\ y &= \frac{1}{2}x + 2 \end{align*} }$

This system consists of two linear equations with two variables, x and y. The first equation is:

${ x + 2y = 6 }$

And the second equation is:

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

Substitution Method

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

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

${ x + 2(\frac{1}{2}x + 2) = 6 }$

Simplifying the Equation

To simplify the equation, we can start by distributing the 2 to the terms inside the parentheses:

${ x + x + 4 = 6 }$

This simplifies to:

${ 2x + 4 = 6 }$

Solving for x

To solve for x, we can subtract 4 from both sides of the equation:

${ 2x = 2 }$

Now, we can divide both sides of the equation by 2:

${ x = 1 }$

Substituting x into the Second Equation

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

${ y = \frac{1}{2}(1) + 2 }$

This simplifies to:

${ y = \frac{1}{2} + 2 }$

Simplifying the Expression

To simplify the expression, we can start by combining the fractions:

${ y = \frac{1}{2} + \frac{4}{2} }$

This simplifies to:

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

The Result

As we can see, the system of equations has a unique solution, which is x = 1 and y = 5/2.

Geometric Interpretation

Geometrically, the system of equations can be represented as a pair of lines on a coordinate plane. The first equation represents a line with a slope of 2 and a y-intercept of 3. The second equation represents a line with a slope of 1/2 and a y-intercept of 2. Since the two lines intersect at a single point, the system of equations has a unique solution.

Conclusion

In conclusion, the system of equations has a unique solution, which is x = 1 and y = 5/2. This is because the two equations represent lines that intersect at a single point.

Example 4

Consider the system of equations:

${ \begin{align*} x + 2y &= 6 \\ y &= -\frac{1}{2}x + 3 \end{align*} }$

This system consists of two linear equations with two variables, x and y. The first equation is:

${ x + 2y = 6 }$

And the second equation is:

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

Substitution Method

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

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

${ x + 2(-\frac{1}{2}x + 3) = 6 }$

Simplifying the Equation

To simplify the equation, we can start by distributing the 2 to the terms inside the parentheses:

${ x - x + 6 = 6 }$

This simplifies to:

${ 6 = 6 }$

The Result

As we can see, the equation is an identity, which means that it is true for all values of x. This means that the system of equations has infinitely many solutions.

Geometric Interpretation

Conclusion

In conclusion, the system of equations has no

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Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system.

Q: How do I solve a system of linear equations?

A: There are several methods to solve a system of linear equations, including the substitution method, the elimination method, and the graphing method. The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one of the variables. The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection.

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

A: A system of linear equations consists of linear equations, which are equations that can be written in the form ax + by = c, where a, b, and c are constants. A system of nonlinear equations, on the other hand, consists of nonlinear equations, which are equations that cannot be written in the form ax + by = c. Nonlinear equations can be quadratic, cubic, or higher-degree equations.

Q: How do I determine the number of solutions to a system of linear equations?

A: To determine the number of solutions to a system of linear equations, you can use the following methods:

If the two equations are identical, the system has infinitely many solutions.
If the two equations are parallel, the system has no solution.
If the two equations intersect at a single point, the system has a unique solution.

Q: Can a system of linear equations have more than one solution?

A: No, a system of linear equations cannot have more than one solution. If a system of linear equations has a solution, it is a unique solution.

Q: Can a system of linear equations have no solution?

A: Yes, a system of linear equations can have no solution. This occurs when the two equations are parallel and do not intersect.

Q: Can a system of linear equations have infinitely many solutions?

A: Yes, a system of linear equations can have infinitely many solutions. This occurs when the two equations are identical and represent the same line.

Q: How do I graph a system of linear equations?

A: To graph a system of linear equations, you can use the following steps:

Graph the first equation on a coordinate plane.
Graph the second equation on the same coordinate plane.
Find the point of intersection between the two lines.

Q: What is the importance of solving systems of linear equations?

A: Solving systems of linear equations is important in many fields, including mathematics, science, engineering, and economics. It is used to model real-world problems, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits.

Q: Can I use technology to solve systems of linear equations?

A: Yes, you can use technology to solve systems of linear equations. Many graphing calculators and computer algebra systems can solve systems of linear equations quickly and accurately.

Q: What are some common mistakes to avoid when solving systems of linear equations?

A: Some common mistakes to avoid when solving systems of linear equations include:

Not checking the work for errors.
Not using the correct method for solving the system.
Not checking the solution to see if it satisfies both equations.

Q: How do I check my work when solving a system of linear equations?

A: To check your work when solving a system of linear equations, you can use the following steps:

Plug the solution into both equations to see if it satisfies both equations.
Check to see if the solution is a unique solution or if it is an identity.
Check to see if the solution is a point of intersection or if it is a line.