Solving Word Problems Using Systems Of Equations

Solving Word Problems Using Systems Of Equations-82
When there is only one solution, the system is called independent, since they cross at only one point.When equations have infinite solutions, they are the same equation, are consistent, and are called dependent or coincident (think of one just sitting on top of the other).

When there is only one solution, the system is called independent, since they cross at only one point.When equations have infinite solutions, they are the same equation, are consistent, and are called dependent or coincident (think of one just sitting on top of the other).

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Push \(Y=\) and enter the two equations in \(=\) and \(=\), respectively. When you get the answer for \(j\), plug this back in the easier equation to get \(d\): \(\displaystyle d=-(4) 6=2\). You’ll want to pick the variable that’s most easily solved for.

Note that we don’t have to simplify the equations before we have to put them in the calculator. You may need to hit “ZOOM 6” (Zoom Standard) and/or “ZOOM 0” (Zoom Fit) to make sure you see the lines crossing in the graph. Let’s try another substitution problem that’s a little bit different: Probably the most useful way to solve systems is using linear combination, or linear elimination.

So the points of intersections satisfy both equations simultaneously.

We’ll need to put these equations into the \(y=mx b\) (\(d=mj b\)) format, by solving for the \(d\) (which is like the \(y\)): First of all, to graph, we had to either solve for the “\(y\)” value (“\(d\)” in our case) like we did above, or use the cover-up, or intercept method.

The easiest way for the second equation would be the intercept method; when we put for the “\(d\)” intercept.

We can do this for the first equation too, or just solve for “\(d\)”.Now let’s see why we can add, subtract, or multiply both sides of equations by the same numbers – let’s use real numbers as shown below.Remember these are because of the Additive Property of Equality, Subtraction Property of Equality, Multiplicative Property of Equality, and Division Property of Equality: \(\displaystyle \begin\color\\\,\left( \right)\left( \right)=\left( \right)6\text\\,\,\,\,-25j-25d\,=-150\,\\,\,\,\,\,\underline\text\\,\,\,0j 25d=\,50\\25d\,=\,50\d=2\\d j\,\,=\,\,6\\,2 j=6\j=4\end\). \(\displaystyle \begin\color\,\,\,\,\,\,\,\text-3\\color\text\,\,\,\,\,\,\,\text5\end\) \(\displaystyle \begin-6x-15y=3\,\\,\underline\text\\,29x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=58\\,\,\,\,\,\,\,\,\,\,\,\,\,x=2\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\2(2) 5y=-1\\,\,\,\,\,\,4 5y=-1\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,5y=-5\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y=-1\end\) ). In the example above, we found one unique solution to the set of equations.When equations have no solutions, they are called inconsistent equations, since we can never get a solution.Here are graphs of inconsistent and dependent equations that were created on the graphing calculator: Let’s get a little more complicated with systems; in real life, we rarely just have two unknowns with two equations.Note that we solve Algebra Word Problems without Systems here, and we solve systems using matrices in the Matrices and Solving Systems with Matrices section here.“Systems of equations” just means that we are dealing with more than one equation and variable.Always write down what your variables will be: equations as shown below.Notice that the \(j\) variable is just like the \(x\) variable and the \(d\) variable is just like the \(y\).(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).The main purpose of the linear combination method is to add or subtract the equations so that one variable is eliminated.

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