Problem 88 - Algebra and Trigonometry

Question

Explain how to solve a system of equations using the addition method. Use \(3 x+5 y=-2\) and \(2 x+3 y=0\) to illustrate your explanation.

Step-by-Step Solution

Answer

The solution to the system of the given equations is \(x = 6\) and \(y = -4\).

Step 1: Multiply the Equations

Modifying the given equations to make the coefficients of 'y' in both the equations same which can cancel out each other. Selecting the variable 'y' to eliminate, Multiply the first equation by 3 and the second equation by 5, to make the coefficients of y the same in both equations. So, the equations become: \(9x + 15y = -6\) (equation 1) and \(10x + 15y = 0\) (equation 2)

Step 2: Add the Equations

Add the two equations together. The addition of the two equations eliminates the 'y'. So, the equation now becomes: \(-x = -6\).

Step 3: Solve for the remaining variable

Divide both sides by -1 to solve for 'x'. So, \(x = 6\).

Step 4: Substitute x = 6 into one of the original equations

Select one of the original equations to substitute 'x' value into. For example, the original equation \(2x + 3y = 0\) becomes: \(2(6) + 3y = 0\). Which simplifies to: \(12 + 3y = 0\).

Step 5: Solve for 'y'

Subtract 12 from both sides: \(3y = -12\). Divide both sides by 3 to solve for 'y'. So, \(y = -4\).