Solving Systems By Substitution Word Problems

Solving Systems By Substitution Word Problems-1
(You can also use the WINDOW button to change the minimum and maximum values of your \(x\) and \(y\) values.) TRACE” (CALC), and then either push 5, or move cursor down to intersect. The reason it’s most useful is that usually in real life we don’t have one variable in terms of another (in other words, a “\(y=\)” situation).

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Then we add the two equations to get “\(0j\)” and eliminate the “\(j\)” variable (thus, the name “linear elimination”). Now that we get \(d=2\), we can plug in that value in the either original equation (use the easiest! We then get the second set of equations to add, and the \(y\)’s are eliminated. Now we can plug in that value in either original equation (use the easiest! Sometimes, however, there are no solutions (when lines are parallel) or an infinite number of solutions (when the two lines are actually the same line, and one is just a “multiple” of the other) to a set of equations.

When there is at least one solution, the equations are consistent equations, since they have a solution.

So far, we’ve basically just played around with the equation for a line, which is \(y=mx b\).

Now, you can always do “guess and check” to see what would work, but you might as well use algebra!

So, again, now we have three equations and three unknowns (variables).

We’ll learn later how to put these in our calculator to easily solve using matrices (see the Matrices and Solving Systems with Matrices section), but for now we need to first use two of the equations to eliminate one of the variables, and then use two other equations to eliminate the same variable: Now this gets more difficult to solve, but remember that in “real life”, there are computers to do all this work!

This means that the numbers that work for both equations is 4 pairs of jeans and 2 dresses! Here is the problem again: Solve for \(d\): \(\displaystyle d=-j 6\).

We can also use our graphing calculator to solve the systems of equations: Solve for \(y\,\left( d \right)\) in both equations. Plug this in for \(d\) in the second equation and solve for \(j\). Note that we could have also solved for “\(j\)” first; it really doesn’t matter.

It’s easier to put in \(j\) and \(d\) so we can remember what they stand for when we get the answers.

There are several ways to solve systems; we’ll talk about graphing first.


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