6 Questions
What is the primary purpose of the substitution method in solving linear equations in two variables?
To solve for one variable and substitute that value into the other equation
In the system of equations 2x + y = 7 and x - 2y = 6, what is the value of y when substituted into the second equation using the substitution method?
-3.2
When using the elimination method to solve a system of linear equations, what is a key condition for its application?
At least one equation should be in standard form (Ax + By = C)
What is the key strategy behind the elimination method in solving linear equations?
Adding or subtracting the equations to eliminate one variable
If a system of linear equations is given as 3x - 2y = 5 and 4x + y = 8, which method would be more suitable for solving this system?
Elimination method
What should be done if both equations of a linear system are not in standard form for the elimination method to apply directly?
Rearrange one equation to match standard form
Study Notes
Solving Linear Equations in Two Variables: Substitution and Elimination Methods
Solving linear equations in two variables involves finding the values of x and y that satisfy both equations simultaneously. We can use two main methods to solve systems of linear equations: the substitution method and the elimination method.
Substitution Method
The substitution method is used when one of the equations is already solved for one of the variables. In this case, we can solve for that variable and then substitute the value into the other equation to solve for the remaining variable.
For example, consider the system of equations:
- 2x + y = 7
- x - 2y = 6
We can solve the first equation for y:
2x + y = 7 y = 7 - 2x
Now, we can substitute y in the second equation:
x - 2(7 - 2x) = 6 x - 14 + 4x = 6
Combining like terms, we get:
5x = 12
Dividing by 5, we find:
x = 2.4
Now, we can substitute x back into the first equation to solve for y:
2(2.4) + y = 7 y = 7 - 4.8 = -3.2
The solution is (2.4, -3.2).
Elimination Method
The elimination method is used when both equations are written in the standard form (Ax + By = C) or when one equation is in the standard form and the other is in the form of Ax + By = C or By = 3y.
For example, consider the system of equations:
- 3x + 2y = 20
- 4x - 2y = 12
We can eliminate y by multiplying the first equation by 2 and the second equation by 2:
3x + 2y = 20 6x + 4y = 40 4x - 2y = 12 8x - 4y = 24
Now, we can add the two equations:
10x - 4y = 64
Dividing by 4, we find:
x = 6.4
Now, we can substitute x back into one of the equations to solve for y:
3(6.4) + 2y = 20 y = 20 - 19.2 = 0.8
The solution is (6.4, 0.8).
Choosing the Most Convenient Method
In general, the choice of method depends on the form of the equations. If one equation is already solved for one of the variables, the substitution method is more convenient. If both equations are in the standard form, the elimination method is more convenient.
Learn how to solve systems of linear equations in two variables using the substitution and elimination methods. Understand when to use each method and follow step-by-step examples to find solutions for x and y.
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