Elimination Method

Summary

This video explains the elimination method for solving systems of equations, focusing on eliminating variables to find solutions for x and y. The tutorial demonstrates several examples, showing how to rearrange equations, manipulate coefficients, and apply basic algebraic operations to isolate variables. Viewers are guided through each step with clear explanations, including how to handle both positive and negative coefficients. The video also encourages viewers to practice solving similar problems for a better understanding of the process.

Takeaways

• 😀 The elimination method is a technique for solving systems of equations by eliminating one variable to solve for the other.
• 😀 It's not always necessary to align the variables, as the equations can be rearranged depending on the given problem.
• 😀 The goal is to eliminate one variable by either adding or subtracting the equations, making it easier to solve for the remaining variable.
• 😀 Once one variable is found, substitute it back into any of the original equations to solve for the other variable.
• 😀 In some cases, it's necessary to multiply both sides of an equation by a constant to create opposite coefficients for elimination.
• 😀 When coefficients are already opposites (e.g., +6y and -6y), simply add or subtract the equations to eliminate the variable.
• 😀 After finding one variable, solving for the other can be done by plugging the known value into any of the original equations.
• 😀 The process involves careful manipulation of equations, like subtracting terms from both sides or multiplying to achieve desired coefficients.
• 😀 The solution to the system of equations is the pair of values (x, y) that satisfy both equations simultaneously.
• 😀 The video emphasizes practicing the method and pausing to attempt problems independently to ensure understanding.

Q & A

What is the main concept behind the elimination method of solving systems of equations?

The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the other variable.

When using the elimination method, is it always necessary to align the variables and constants in the equations?

No, it is not always necessary to align the variables and constants. In some cases, rearranging terms might be required, but alignment is not a strict rule.

In the example with 4x + 6y = 32, how do you solve for y once x is found?

After finding x (which is 5), substitute it into the equation 4x + 6y = 32. This gives 4(5) + 6y = 32, which simplifies to 20 + 6y = 32. Subtract 20 from both sides to get 6y = 12, then divide by 6 to find y = 2.

Why is it necessary to multiply an equation by a constant, such as multiplying by -1, in some cases during elimination?

Multiplying by a constant, like -1, is done to create opposite coefficients for one of the variables. This allows you to eliminate that variable when you add or subtract the equations.

How do you solve for x once you have found y in a system of equations?

To solve for x, substitute the value of y into one of the original equations, and then solve the resulting equation for x.

In the example where x - 2y = -10 and y = 7, how do you solve for x?

Substitute y = 7 into the equation x - 2y = -10. This gives x - 2(7) = -10, which simplifies to x - 14 = -10. Add 14 to both sides to get x = 4.

What do you do when the coefficients of the variables in the system do not directly oppose each other?

When the coefficients do not oppose each other, you may need to multiply one or both equations by a constant to create opposite coefficients, allowing you to eliminate one variable.

In the problem with 7x + 18y = 25 and -2x + 6y = 12, how do you eliminate x or y?

To eliminate x, multiply the second equation by 3 to get 9x + 18y = 36. Then subtract the first equation (7x + 18y = 25) from this, which results in 2x = 11. Then solve for x.

In the case of 6x + y = -39 and y = 3, how do you solve for x?

Substitute y = 3 into the equation 6x + y = -39. This gives 6x + 3 = -39. Subtract 3 from both sides to get 6x = -42, then divide by 6 to find x = -7.

What does it mean when the solution to a system of equations is expressed as a fraction, such as x = -5/4?

Expressing a solution as a fraction, like x = -5/4, indicates that the exact solution cannot be simplified into a whole number. The fraction is the most precise form of the solution.