Solving simultaneous equations can feel like solving a complex puzzle, especially when you have multiple equations with the same variables. But fear not! Understanding how to crack these equations is easier than you might think. In this blog, we'll delve into the key methods for solving simultaneous equations, such as substitution, elimination, and graphical approaches.
What Are Simultaneous Equations?
Simultaneous equations are a set of equations that share the same unknown variables. A solution to these equations is a set of values for the variables that satisfy all the equations at the same time. For example:
2x - 4y = 4, 5x + 8y = 3
2a - 3b + c = 9, a + b + c = 2, a - b - c = 9
3x - y = 5, x - y = 4
a2 + b2 = 9, a2 - b2 = 16
We can solve such a set of equations using different methods. Let us discuss different methods to solve simultaneous equations in the next section.
We can solve these equations using different methods. We'll discuss these methods in the next section.
Methods for Solving Simultaneous Equations
- Substitution Method
- Elimination Method
- Graphical Method
Depending on the equations, there can be no solution, one unique solution, or infinitely many solutions. Additional methods, like cross multiplication and determinants, can also be used for linear equations in two variables.
To solve simultaneous equations, you need as many equations as there are unknown variables. We’ll cover these methods with examples in the following sections.
Rules To Apply Simultaneous EquationsÂ
To solve simultaneous equations, we follow certain rules first to simplify the equations. Some of the important rules are:
- Simplify each equation by removing any parentheses.
- Combine like terms.
- Isolate the variable terms on one side of the equation.
- Use the chosen method to solve for the variables.
Solving Simultaneous Equations Using Substitution Method
Now that we have discussed different methods to solve simultaneous equations. Let us solve a few examples using the substitution method to understand it better. Consider a system of equations x + y = 4 and 2x - 3y = 9. Now, we will find the value of one variable in terms of another variable using one of the equations and substitute it into the other equation. We have
x + y = 4 --- (1)
2x - 3y = 9 --- (2)
From (1), we have
x = 4 - y --- (3)
Substituting this in (2), we get
2(4 - y) - 3y = 9
⇒ 8 - 2y - 3y = 9
⇒ 8 - 5y = 9
Isolating the variable term to one side of the equation, we have
⇒ -5y = 9 - 8
⇒ y = 1/(-5)
= -1/5
Substituting the value of y in (3), we have
x = 4 - (-1/5)
= 4 + 1/5
= (20 + 1)/5
= 21/5
Answer: So, the solution of the simultaneous equations x + y = 4 and 2x - 3y = 9 is x = 21/5 and y = -1/5.
Solving Simultaneous Equations By Elimination Method
To solve simultaneous equations by the elimination method, we eliminate a variable from one equation using another to find the value of the other variable. Let us solve an example to understand find the solution of simultaneous equations using the elimination method. Consider equations 2x - 5y = 3 and 3x - 2y = 5. We have
2x - 5y = 3 --- (1)
and 3x - 2y = 5 --- (2)
Here, we will eliminate the variable y, so we find the LCM of the coefficients of y. LCM (5, 2) = 10. So, multiply equation (1) by 2 and equation (2) by 5. So, we have
[ 2x - 5y = 3 ] × 2
⇒ 4x - 10y = 6 --- (3)
[ 3x - 2y = 5 ] × 5
⇒ 15x - 10y = 25 --- (4)
Now, subtracting equation (3) from (4), we have
(15x - 10y) - (4x - 10y) = 25 - 6
⇒ 15x - 10y - 4x + 10y = 19
⇒ (15x - 4x) + (-10y + 10y) = 19
⇒ 11x + 0 = 19
⇒ x = 19/11
Now, substituting this value of x in (1), we have
2(19/11) - 5y = 3
⇒ 38/11 - 5y = 3
⇒ 5y = 38/11 - 3
⇒ 5y = (38 - 33) / 11
⇒ y = 5/(11×5)
= 1/11
So, the solution of the simultaneous equations 2x - 5y = 3 and 3x - 2y = 5 using the elimination method is x = 19/11 and y = 1/11.
Solving Simultaneous Equations Graphically
In this section, we will learn to solve the simultaneous equations using the graphical method. We will plot the lines on the coordinate plane and then find the point of intersection of the lines to find the solution. Consider simultaneous equations x + y = 10 and x - y = 4. Now, find two points (x, y) satisfying for each equation such that the equation holds.
For x + y = 10, we have
So, we have coordinates (0, 10) and (10, 0). Plot them and join the points and plot the line x + y = 10.
For equation x - y = 4, we have
x - y = 4
So, we have coordinates (0, -4) and (4, 0). Plot them and join the points and plot the line x - y = 4.
Solving simultaneous equations
Now, as we have plotted the two lines, find their intersecting point. The two lines x + y = 10 and x - y = 4 intersect each other at (7, 3). So, we have found the solution of the simultaneous equations x + y = 10 and x - y = 4 graphically which is x = 7 and y = 3.
Important Notes on Simultaneous Equations
- Simultaneous equations are solved together because they share common variables.
- Methods to solve them include substitution, elimination, and graphical methods.
- Cross multiplication and determinant methods are also useful for linear equations in two variables
Study Algebra with MTS
Mastering simultaneous equations might seem daunting, but fear not! With the right tools and some practice, you'll conquer these problems like a pro. Whether you're into substitution, elimination, or even the graphical approach, nailing these techniques will boost your math skills and open up new opportunities for growth.
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