Solve The System Of Linear Equations:$\[ \begin{align*} y &= -\frac{7}{4}x + \frac{5}{2} \\ y &= \frac{3}{4}x - 3 \end{align*} \\]Billy Graphed These Equations To Find Their Intersection Point.

Introduction

In mathematics, a system of linear equations is a set of two or more linear 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 with two variables. We will use the given equations to find their intersection point, which is the solution to the system.

The Given Equations

The given system of linear equations is:

$$\begin{align*} y &= -\frac{7}{4}x + \frac{5}{2} \\ y &= \frac{3}{4}x - 3 \end{align*}$$

Understanding the Equations

To solve the system of linear equations, we need to understand the concept of linear equations. A linear equation is an equation in which the highest power of the variable(s) is 1. In this case, we have two linear equations with two variables, x and y.

Equation 1: $y = -\frac{7}{4}x + \frac{5}{2}$

The first equation is $y = -\frac{7}{4}x + \frac{5}{2}$. This equation represents a line with a slope of $-\frac{7}{4}$ and a y-intercept of $\frac{5}{2}$.

Equation 2: $y = \frac{3}{4}x - 3$

The second equation is $y = \frac{3}{4}x - 3$. This equation represents a line with a slope of $\frac{3}{4}$ and a y-intercept of $-3$.

Finding the Intersection Point

To find the intersection point of the two lines, we need to set the two equations equal to each other and solve for x. This is because the intersection point is the point where the two lines intersect, and at this point, the values of x and y are the same for both equations.

Setting the Equations Equal to Each Other

We set the two equations equal to each other:

$$-\frac{7}{4}x + \frac{5}{2} = \frac{3}{4}x - 3$$

Simplifying the Equation

To simplify the equation, we can multiply both sides by 4 to eliminate the fractions:

$$-7x + 10 = 3x - 12$$

Solving for x

Now, we can solve for x by isolating the variable x on one side of the equation. We can do this by adding 7x to both sides of the equation and then adding 12 to both sides:

$$-7x + 7x + 10 = 3x + 7x - 12$$

$$10 = 10x - 12$$

$$10 + 12 = 10x - 12 + 12$$

$$22 = 10x$$

Dividing Both Sides by 10

Finally, we can divide both sides of the equation by 10 to solve for x:

$$\frac{22}{10} = \frac{10x}{10}$$

$$\frac{11}{5} = x$$

Finding the Value of y

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

$$y = -\frac{7}{4}x + \frac{5}{2}$$

Substituting x = $\frac{11}{5}$ into the equation, we get:

$$y = -\frac{7}{4}\left(\frac{11}{5}\right) + \frac{5}{2}$$

$$y = -\frac{77}{20} + \frac{25}{10}$$

$$y = -\frac{77}{20} + \frac{50}{20}$$

$$y = -\frac{27}{20}$$

Conclusion

In this article, we solved a system of two linear equations with two variables. We used the given equations to find their intersection point, which is the solution to the system. We set the two equations equal to each other, simplified the equation, solved for x, and then found the value of y by substituting the value of x into one of the original equations. The intersection point of the two lines is $\left(\frac{11}{5}, -\frac{27}{20}\right)$.

Final Answer

$$

Introduction

In our previous article, we solved a system of two linear equations with two variables. We used the given equations to find their intersection point, which is the solution to the system. In this article, we will provide a Q&A guide to help you understand the concept of solving systems of linear equations.

Q: What is a system of linear equations?

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

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

A: A system of linear equations has a solution if the lines represented by the equations intersect at a single point. If the lines are parallel, the system has no solution.

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, while a system of nonlinear equations consists of nonlinear equations. Nonlinear equations are equations in which the highest power of the variable(s) is not 1.

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

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

1. Set the two equations equal to each other.
2. Simplify the equation.
3. Solve for x.
4. Find the value of y by substituting the value of x into one of the original equations.

Q: What is the intersection point of two lines?

A: The intersection point of two lines is the point where the two lines intersect. At this point, the values of x and y are the same for both equations.

Q: How do I find the intersection point of two lines?

A: To find the intersection point of two lines, you can set the two equations equal to each other, simplify the equation, solve for x, and then find the value of y by substituting the value of x into one of the original equations.

Q: What is the difference between a dependent and an independent system of linear equations?

A: A dependent system of linear equations is a system in which the equations are not independent. In other words, one equation is a multiple of the other. An independent system of linear equations is a system in which the equations are independent.

Q: How do I determine if a system of linear equations is dependent or independent?

A: To determine if a system of linear equations is dependent or independent, you can check if one equation is a multiple of the other. If one equation is a multiple of the other, the system is dependent. If the equations are not multiples of each other, the system is independent.

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

A: Solving systems of linear equations is significant in many areas of mathematics and science, including algebra, geometry, trigonometry, and physics. It is used to model real-world problems and to solve problems in fields such as engineering, economics, and computer science.

Conclusion

In this article, we provided a Q&A guide to help you understand the concept of solving systems of linear equations. We covered topics such as the definition of a system of linear equations, how to determine if a system has a solution, and how to solve a system of linear equations. We also discussed the significance of solving systems of linear equations and how it is used in various fields.

Final Answer

The final answer is $\boxed{\left(\frac{11}{5}, -\frac{27}{20}\right)}$.