Eliminating Variables in Equations

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.