A System Of Equations Is Shown:${ \begin{cases} y = 2x - 7 \ y = -x + 5 \end{cases} }$What Is The Solution To The System Of Equations?

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Introduction

A system of equations is a set of two or more equations that contain the same variables. In this case, we have a system of two linear equations with two variables, x and y. The system is given by:

{ \begin{cases} y = 2x - 7 \\ y = -x + 5 \end{cases} \}

Our goal is to find the solution to this system of equations, which is the point of intersection between the two lines represented by the equations.

Understanding the Equations

The first equation is y = 2x - 7, which is a linear equation in slope-intercept form. The slope of this line is 2, and the y-intercept is -7. This means that the line passes through the point (0, -7) and has a slope of 2.

The second equation is y = -x + 5, which is also a linear equation in slope-intercept form. The slope of this line is -1, and the y-intercept is 5. This means that the line passes through the point (0, 5) and has a slope of -1.

Graphing the Equations

To visualize the system of equations, we can graph the two lines on a coordinate plane. The first line, y = 2x - 7, has a slope of 2 and a y-intercept of -7. We can plot this line by starting at the y-intercept (0, -7) and moving up 2 units for every 1 unit we move to the right.

The second line, y = -x + 5, has a slope of -1 and a y-intercept of 5. We can plot this line by starting at the y-intercept (0, 5) and moving down 1 unit for every 1 unit we move to the right.

Finding the Solution

To find the solution to the system of equations, we need to find the point of intersection between the two lines. This is the point where the two lines cross each other.

We can find the point of intersection by setting the two equations equal to each other and solving for x. This gives us:

2x - 7 = -x + 5

We can add x to both sides of the equation to get:

3x - 7 = 5

Next, we can add 7 to both sides of the equation to get:

3x = 12

Finally, we can divide both sides of the equation by 3 to get:

x = 4

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 - 7.

y = 2(4) - 7 y = 8 - 7 y = 1

Therefore, the solution to the system of equations is (4, 1).

Conclusion

In this article, we have discussed a system of two linear equations with two variables, x and y. We have graphed the two lines on a coordinate plane and found the point of intersection between the two lines, which is the solution to the system of equations. We have used algebraic methods to solve for x and then substituted the value of x into one of the original equations to find the value of y.

Step-by-Step Solution

Here is a step-by-step solution to the system of equations:

  1. Set the two equations equal to each other: 2x - 7 = -x + 5
  2. Add x to both sides of the equation: 3x - 7 = 5
  3. Add 7 to both sides of the equation: 3x = 12
  4. Divide both sides of the equation by 3: x = 4
  5. Substitute the value of x into one of the original equations: y = 2(4) - 7
  6. Simplify the equation: y = 8 - 7
  7. Solve for y: y = 1

Therefore, the solution to the system of equations is (4, 1).

Final Answer

The final answer is (4, 1).

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Introduction

In our previous article, we discussed a system of two linear equations with two variables, x and y. We graphed the two lines on a coordinate plane and found the point of intersection between the two lines, which is the solution to the system of equations. In this article, we will answer some common questions related to systems of equations.

Q&A

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that contain the same variables. In this case, we have a system of two linear equations with two variables, x and y.

Q: How do I graph a system of equations?

A: To graph a system of equations, you can plot the two lines on a coordinate plane. The first line, y = 2x - 7, has a slope of 2 and a y-intercept of -7. The second line, y = -x + 5, has a slope of -1 and a y-intercept of 5.

Q: How do I find the solution to a system of equations?

A: To find the solution to a system of equations, you can set the two equations equal to each other and solve for x. Then, substitute the value of x into one of the original equations to find the value of y.

Q: What if the two lines are parallel?

A: If the two lines are parallel, they will never intersect, and there will be no solution to the system of equations.

Q: What if the two lines are perpendicular?

A: If the two lines are perpendicular, they will intersect at a single point, and there will be a unique solution to the system of equations.

Q: Can I use substitution or elimination to solve a system of equations?

A: Yes, you can use substitution or elimination to solve a system of equations. Substitution involves solving one equation for one variable and then substituting that expression into the other equation. Elimination involves adding or subtracting the two equations to eliminate one variable.

Q: How do I know which method to use?

A: You can use the following steps to determine which method to use:

  • If the coefficients of one variable are the same in both equations, use elimination.
  • If the coefficients of one variable are different in both equations, use substitution.

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

A: Yes, you can use a graphing calculator to solve a system of equations. Simply graph the two lines and find the point of intersection.

Conclusion

In this article, we have answered some common questions related to systems of equations. We have discussed how to graph a system of equations, how to find the solution to a system of equations, and how to use substitution or elimination to solve a system of equations. We have also discussed how to use a graphing calculator to solve a system of equations.

Final Answer

The final answer is (4, 1).

Additional Resources

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