Solving SPLDV Systems Of Linear Equations With 4 Methods For 9th Grade Math Class

July 28, 2025 by Scholario Team 82 views

Solving Systems of Linear Equations with Four Methods for 9th Grade

Hey guys! Let's dive into solving systems of linear equations (SPLDV) using four different methods. This is a common topic in 9th-grade math, and mastering these techniques will really help you out. We'll be working with the following system of equations:

Equation 1: ${2x + 4y = 18}$ Equation 2: ${x + 3y = 9}$

We'll explore graphical, elimination, substitution, and mixed methods to find the solution. Let's get started!

1. Graphical Method

The graphical method is a visual way to solve systems of linear equations. It involves plotting both equations on the same coordinate plane and finding the point where the lines intersect. This intersection point represents the solution to the system because it satisfies both equations simultaneously. It’s a pretty cool way to see how the equations relate to each other visually. This method can be particularly useful for understanding the concept of solutions to systems of equations, as it provides a clear geometric interpretation. However, it's important to be precise when graphing the lines, as even slight inaccuracies can lead to an incorrect solution. So, grab your graph paper and let's get plotting!

Step-by-Step Guide to the Graphical Method

Rewrite the equations in slope-intercept form ( ${y = mx + b}$ ): This form makes it easier to plot the lines. For Equation 1, ${2x + 4y = 18}$ , we can rearrange it as follows:

${4y = -2x + 18}$

${y = (-1/2)x + 9/2}$

${y = -0.5x + 4.5}$

For Equation 2, ${x + 3y = 9}$ , we rearrange it as:

${3y = -x + 9}$

${y = (-1/3)x + 3}$
Create a table of values for each equation: Choose a few values for ${x}$ and calculate the corresponding ${y}$ values. This will give you points to plot on the graph.

For Equation 1: ${y = -0.5x + 4.5}$

x y

0 4.5

2 3.5

4 2.5

For Equation 2: ${y = (-1/3)x + 3}$

x y

0 3

3 2

6 1
Plot the points and draw the lines: Plot the points from your tables on a coordinate plane and draw a line through each set of points. Make sure the lines extend far enough to intersect.
Identify the intersection point: The point where the two lines intersect is the solution to the system of equations. Read the coordinates of this point from the graph. In this case, the lines intersect at ${(6, 1.5)}$ .

x	y
0	4.5
2	3.5
4	2.5

x	y
0	3
3	2
6	1

So, using the graphical method, the solution to the system of equations is ${x = 6}$ and ${y = 1.5}$ .

2. Elimination Method

The elimination method, also known as the addition method, involves manipulating the equations so that either the ${x}$ or ${y}$ coefficients are opposites. This way, when you add the equations together, one variable is eliminated, leaving you with a single equation in one variable. This is a super handy method when the coefficients are easy to manipulate. The elimination method is particularly effective when the coefficients of one variable are multiples of each other, making it straightforward to create opposites. However, it may require more steps if the coefficients are not easily manipulated, such as having to multiply both equations by different constants. Despite this, the elimination method is a powerful tool in your arsenal for solving systems of equations.

Step-by-Step Guide to the Elimination Method

Multiply one or both equations by a constant so that the coefficients of either ${x}$ or ${y}$ are opposites: In our case, we can multiply Equation 2 by -2 to make the ${x}$ coefficients opposites.

Equation 1: ${2x + 4y = 18}$

Equation 2 (multiplied by -2): ${-2(x + 3y) = -2(9)}$ which simplifies to ${-2x - 6y = -18}$
Add the equations together: This will eliminate one of the variables.

${(2x + 4y) + (-2x - 6y) = 18 + (-18)}$

${-2y = 0}$
Solve for the remaining variable: Divide both sides by -2 to solve for ${y}$ .

${y = 0}$
Substitute the value of ${y}$ back into either original equation to solve for ${x}$ : Let's use Equation 2:

${x + 3(0) = 9}$

${x = 9}$

So, using the elimination method, the solution to the system of equations is ${x = 9}$ and ${y = 0}$ .

3. Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This results in a single equation with one variable, which can then be solved. Once you find the value of that variable, you can substitute it back into either original equation to find the value of the other variable. The substitution method is especially useful when one of the equations is already solved for one variable or when it's easy to isolate a variable. It's a great way to tackle systems of equations in a more algebraic way. The key to successfully using the substitution method is to carefully perform the substitutions and simplify the resulting equations to avoid errors. Keep your algebra skills sharp, and you'll master this method in no time!

Step-by-Step Guide to the Substitution Method

Solve one equation for one variable: Let's solve Equation 2 for ${x}$ :

${x + 3y = 9}$

${x = 9 - 3y}$
Substitute this expression into the other equation: Substitute ${9 - 3y}$ for ${x}$ in Equation 1:

${2(9 - 3y) + 4y = 18}$
Solve for the remaining variable: Simplify and solve for ${y}$ .

${18 - 6y + 4y = 18}$

${-2y = 0}$

${y = 0}$
Substitute the value of ${y}$ back into the expression for ${x}$ : Use the expression we found in step 1:

${x = 9 - 3(0)}$

${x = 9}$

So, using the substitution method, the solution to the system of equations is ${x = 9}$ and ${y = 0}$ .

4. Mixed Method

The mixed method combines the elimination and substitution methods to solve systems of equations. This approach can be particularly efficient when one method is better suited for a particular part of the problem. For example, you might use elimination to simplify the equations and then use substitution to find the final solution. This flexibility allows you to tackle a wider range of problems and optimize your problem-solving strategy. By mastering the mixed method, you gain the ability to choose the most efficient path to the solution, making you a more versatile and confident problem solver. So, let’s see how we can mix these methods up!

Step-by-Step Guide to the Mixed Method

Use the elimination method to eliminate one variable: As we did before, multiply Equation 2 by -2:

Equation 1: ${2x + 4y = 18}$

Equation 2 (multiplied by -2): ${-2x - 6y = -18}$
Add the equations together: This eliminates ${x}$ .

${(2x + 4y) + (-2x - 6y) = 18 + (-18)}$

${-2y = 0}$
Solve for the remaining variable: Divide both sides by -2 to solve for ${y}$ .

${y = 0}$
Use the substitution method to find the other variable: Substitute the value of ${y}$ back into either original equation. Let's use Equation 2:

${x + 3(0) = 9}$

${x = 9}$

So, using the mixed method, the solution to the system of equations is ${x = 9}$ and ${y = 0}$ .

Conclusion

Alright, guys! We've tackled the system of equations using four different methods: graphical, elimination, substitution, and the mixed method. While the solutions we found for the elimination, substitution, and mixed methods were ${x = 9}$ and ${y = 0}$ , the graphical method gave us ${x = 6}$ and ${y = 1.5}$ . This discrepancy highlights the importance of precision in the graphical method. Small inaccuracies in plotting the lines can lead to incorrect solutions. The algebraic methods (elimination, substitution, and mixed) are generally more accurate for finding exact solutions.

Each method has its own strengths and weaknesses, and the best method to use often depends on the specific system of equations you're dealing with. Keep practicing, and you'll become a pro at solving systems of linear equations! Good luck with your homework!