Mastering Linear Simultaneous Equations: A Step-by-Step Guide

In mathematics, a system of linear simultaneous equations is a set of two or more equations with two or more variables, which need to be solved simultaneously. A solution to this system is a set of values for each variable that satisfies all of the equations in the system. In this study guide, we will explain the concept of solutions of linear simultaneous equations and provide examples to help students understand this topic more easily.

Solving a System of Linear Simultaneous Equations:

There are different methods for solving a system of linear simultaneous equations, including substitution, elimination, and matrix methods. We will explain the substitution and elimination methods, as they are often used in introductory algebra courses.

Substitution Method:

The substitution method involves solving one equation for one variable and substituting the resulting expression into the other equation(s). Here's an example to illustrate the substitution method:

Find the solution to the system of equations:
3x + 2y = 11
x - y = 1

Step 1: Solve one equation for one variable.
x = y + 1

Step 2: Substitute the expression obtained in step 1 into the other equation.
3(y+1) + 2y = 11

Step 3: Solve for the remaining variable.
5y = 8
y = 8/5

Step 4: Substitute the value of y into one of the original equations to find the value of x.
x = y + 1 = 8/5 + 1 = 13/5

Therefore, the solution to the system of equations is (x, y) = (13/5, 8/5).

Elimination Method:

Find the solution to the system of equations:
3x + 2y = 11
x - y = 1

Step 1: Solve one equation for one variable.
x = y + 1

Step 2: Substitute the expression obtained in step 1 into the other equation.
3(y+1) + 2y = 11

Step 3: Solve for the remaining variable.
5y = 8
y = 8/5

Step 4: Substitute the value of y into one of the original equations to find the value of x.
x = y + 1 = 8/5 + 1 = 13/5

Therefore, the solution to the system of equations is (x, y) = (13/5, 8/5).

Find the solution to the system of equations:
2x - 3y = 5
4x + y = 2

Step 1: Multiply one or both equations by a constant to make the coefficients of one variable same in both of the equation. opposite in sign.
Multiply the first equation by 2 to make the coefficient of x is 4. 

Step 2: Now subtract the equation second from First to eliminate one variable.
4x - 6y = 10
4x + y = 2
-7y = 8
x = -8/7

Step 3: Substitute the value of x into one of the original equations to find the value of y.
4(-8/7)+y=2
y=2+32/7
y=46/7

Therefore, the solution to the system of equations is (x, y) = (-8/7, 46/7)

Solving a system of linear simultaneous equations involves finding a set of values for each variable that satisfy all of the equations in the system. The substitution and elimination methods are commonly used to solve these systems. It is important to check the solution to ensure that it satisfies all of the original equations in the system.

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Conclusion

This guide explored the concept of solving linear simultaneous equations, equipping you with the substitution and elimination methods. Remember, practice builds mastery! Don't hesitate to seek personalized tutoring from LearnPick's expert math tutors. With their tailored guidance, you'll conquer these equations and elevate your mathematical understanding.