Question

Describe a general method for deciding which variable to eliminate when using linear combinations.

Short answer

To decide which variable to eliminate when using linear combinations, first inspect the coefficients of the variables in all equations. Chose to eliminate the variable which has the same or opposite coefficients. If no such variable is readily available, modify the coefficients through multiplication or division to make them match either positively or negatively. Then, conduct the linear combination process, solve for the remaining variable, and substitute it back to find the value of the eliminated variable.

Step-by-step solution

Inspect Coefficients

Look at the coefficients of the variables in each equation. If there are any variables with the same or opposite coefficients in different equations, those variables are potential candidates for elimination through addition or subtraction. For example, in the following system of equations: \(2x + 3y = 5\) and \(2x - 3y = 1\), the variable \(x\) has the same coefficients and could be eliminated by subtracting the second equation from the first one.

Modify Coefficients if Necessary

If there are no variables with the same or opposite coefficients, then see if the coefficients of a certain variable could be easily multiplied or divided to make them identical, thus that variable would be a good choice for elimination. For instance, in the system \(3x + 2y = 14\) and \(2x - y = 4\), multiplying the second equation by 2 would result in equivalent coefficients for \(y\), making \(y\) the variable to eliminate.

Eliminate the Selected Variable

Once the targeted variable is decided, proceed to execute the linear combination process. This could involve either adding or subtracting the equations or the modified equations, in order to eliminate the chosen variable. Then, solve the resulting equation to find the value of the remaining variable.

Solve for the Eliminated Variable

Plug the value of the already found variable back into any original equation to solve for the value of the eliminated variable. This gives the complete solution to the system of equations.