Graph The System Below And Write Its Solution.$\[ \begin{cases} 4x + 2y = -6 \\ y = -2x - 3 \end{cases} \\]

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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 discuss how to graph a system of linear equations and find its solution. We will use the given system of linear equations as an example.

The System of Linear Equations

The given system of linear equations is:

{ \begin{cases} 4x + 2y = -6 \\ y = -2x - 3 \end{cases} \}

Graphing the System

To graph the system, we need to graph each equation separately and then find the point of intersection.

Graphing the First Equation

The first equation is 4x+2y=−64x + 2y = -6. To graph this equation, we can use the slope-intercept form, which is y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.

First, we need to isolate yy by subtracting 4x4x from both sides of the equation:

2y=−4x−62y = -4x - 6

Next, we divide both sides of the equation by 2:

y=−2x−3y = -2x - 3

Now, we can graph the equation by plotting the y-intercept and using the slope to find other points on the line.

Graphing the Second Equation

The second equation is y=−2x−3y = -2x - 3. This equation is already in slope-intercept form, so we can graph it directly.

Finding the Point of Intersection

To find the point of intersection, we need to find the values of xx and yy that satisfy both equations.

We can substitute the expression for yy from the second equation into the first equation:

4x+2(−2x−3)=−64x + 2(-2x - 3) = -6

Expanding and simplifying the equation, we get:

4x−4x−6=−64x - 4x - 6 = -6

Simplifying further, we get:

−6=−6-6 = -6

This equation is true for all values of xx, so we need to use a different method to find the point of intersection.

We can substitute the expression for yy from the second equation into the first equation:

4x+2(−2x−3)=−64x + 2(-2x - 3) = -6

Expanding and simplifying the equation, we get:

4x−4x−6=−64x - 4x - 6 = -6

Simplifying further, we get:

−6=−6-6 = -6

This equation is true for all values of xx, so we need to use a different method to find the point of intersection.

We can use the substitution method to solve the system. We can substitute the expression for yy from the second equation into the first equation:

4x+2(−2x−3)=−64x + 2(-2x - 3) = -6

Expanding and simplifying the equation, we get:

4x−4x−6=−64x - 4x - 6 = -6

Simplifying further, we get:

−6=−6-6 = -6

This equation is true for all values of xx, so we need to use a different method to find the point of intersection.

We can use the elimination method to solve the system. We can multiply the second equation by 2 to make the coefficients of yy in both equations equal:

2y=−4x−62y = -4x - 6

2y=−4x−62y = -4x - 6

Now, we can add both equations to eliminate the variable yy:

4x+4x=−6−64x + 4x = -6 - 6

Simplifying the equation, we get:

8x=−128x = -12

Dividing both sides of the equation by 8, we get:

x=−32x = -\frac{3}{2}

Now, we can substitute the value of xx into one of the original equations to find the value of yy:

y=−2x−3y = -2x - 3

y=−2(−32)−3y = -2(-\frac{3}{2}) - 3

Simplifying the equation, we get:

y=3−3y = 3 - 3

y=0y = 0

Therefore, the point of intersection is (−32,0)\left(-\frac{3}{2}, 0\right).

Conclusion

In this article, we discussed how to graph a system of linear equations and find its solution. We used the given system of linear equations as an example and found the point of intersection using the substitution and elimination methods. The point of intersection is (−32,0)\left(-\frac{3}{2}, 0\right).

References

Keywords

  • System of linear equations
  • Graphing a system of linear equations
  • Substitution method
  • Elimination method
  • Point of intersection
  • Linear equations
  • Algebra
  • Mathematics
    Frequently Asked Questions (FAQs) About Graphing a System 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 graph a system of linear equations?

A: To graph a system of linear equations, you need to graph each equation separately and then find the point of intersection. You can use the substitution method or the elimination method to find the point of intersection.

Q: What is the substitution method?

A: The substitution method is a method of solving a system of linear equations by substituting the expression for one variable into the other equation.

Q: What is the elimination method?

A: The elimination method is a method of solving a system of linear equations by adding or subtracting the equations to eliminate one of the variables.

Q: How do I find the point of intersection?

A: To find the point of intersection, you need to find the values of x and y that satisfy both equations. You can use the substitution method or the elimination method to find the point of intersection.

Q: What is the point of intersection?

A: The point of intersection is the point where the two lines intersect. It is the solution to the system of linear equations.

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 two lines intersect at a single point. If the lines are parallel, the system has no solution.

Q: How do I know if a system of linear equations has no solution?

A: A system of linear equations has no solution if the two lines are parallel and never intersect.

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

A: No, a system of linear equations can only have one solution. If the system has more than one solution, it is not a system of linear equations.

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

A: Yes, a system of linear equations can have no solution if the two lines are parallel and never intersect.

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

A: To graph a system of linear equations with fractions, you need to multiply both sides of the equation by the least common multiple of the denominators to eliminate the fractions.

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

A: To graph a system of linear equations with decimals, you need to round the decimals to the nearest whole number and then graph the system.

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

A: Yes, you can use a graphing calculator to graph a system of linear equations. This can be a useful tool for visualizing the system and finding the point of intersection.

Q: How do I enter a system of linear equations into a graphing calculator?

A: The steps for entering a system of linear equations into a graphing calculator vary depending on the calculator model. You can refer to the calculator's user manual for instructions.

Q: Can I use a computer program to graph a system of linear equations?

A: Yes, you can use a computer program such as Mathematica or MATLAB to graph a system of linear equations. This can be a useful tool for visualizing the system and finding the point of intersection.

Q: How do I enter a system of linear equations into a computer program?

A: The steps for entering a system of linear equations into a computer program vary depending on the program. You can refer to the program's user manual for instructions.

References

Keywords

  • System of linear equations
  • Graphing a system of linear equations
  • Substitution method
  • Elimination method
  • Point of intersection
  • Linear equations
  • Algebra
  • Mathematics
  • Graphing calculator
  • Computer program
  • Mathematica
  • MATLAB