Eliminating Variables in Equations

What value does equation 1 (6x - 5y = -4) need to be multiplied by to eliminate the x-terms?

What value does equation 2 (4x + 2y = 28) need to be multiplied by to eliminate the x-terms?

What value does equation 1 (6x - 5y = -4) need to be multiplied by to eliminate the y-terms?

What value does equation 2 (4x + 2y = 28) need to be multiplied by to eliminate the y-terms?

What is the solution for the equations x - y = 12 and x + 2y = 21?

What is the solution for the equations 3x + 4y = -3 and 5x + 3y = 6?

Construct the equations for the statement: The sum of two numbers is 21, and their difference is 9. Equation 1: ______ and Equation 2: ______

Construct the equations for the statement: A certain number is three more than five times another number. Their difference is 11. Equation 1: ______ and Equation 2: ______

How many adult tickets were sold if 200 tickets were sold for $475, with student tickets at $2 each and adult tickets at $3 each?

Eliminating Variables in Equations

To eliminate x-terms in the system consisting of 6x − 5y = -4 and 4x + 2y = 28, multiply Equation 1 by 2 and Equation 2 by -3.
This results in 12x - 10y = -8 and -12x - 6y = -84, effectively cancelling the x-values.

Eliminating y-terms in Equations

To eliminate y-terms in the same equations, multiply Equation 1 by 2 and Equation 2 by 5.
The resulting equations are 12x - 10y = -8 and 20x + 10y = 140, cancelling the y-values.

Solving a System of Equations

For the system x − y = 12 and x + 2y = 21, multiply the first equation by 2 to facilitate cancellation of y-values.
Solve for x and substitute back to find y, yielding the solution (15, 3).

Least Common Multiple in Systems

The system 3x + 4y = -3 and 5x + 3y = 6 requires establishing a least common multiple for the y-values, which is 12.
Multiply the first equation by 3 and the second by 4 to eliminate y, then solve for x and substitute to find y, resulting in the solution (3, -3).

Constructing Equations from Word Problems

When two numbers sum to 21 and have a difference of 9, the equations are x + y = 21 and x - y = 9.
For the scenario where one number is three more than five times another and their difference is 11, the equations are x = 5y + 3 and x - y = 11.

Word Problem Involving Ticket Sales

A scenario with 200 total tickets sold for a concert and a total income of $475 leads to the equations x + y = 200 and 2x + 3y = 475.
Solving for x in the first equation and substituting into the second yields 75 adult tickets sold.

This quiz covers methods for eliminating variables in systems of equations, employing techniques such as multiplication and cancellation. It includes examples demonstrating how to solve systems by eliminating x-terms and y-terms. Test your understanding of these crucial algebra concepts.