Solve each system of equations. If the system has no...

Solve each system of equations. If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{l}\frac{1}{2} x+\frac{1}{3} y=3 \\ \frac{1}{4} x-\frac{2}{3} y=-1\end{array}\right.$

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Key Concepts

System of Linear Equations

This concept involves a set of equations where each equation is linear in the same variables. The aim is to find the values of these variables that simultaneously satisfy all given equations. Solutions to such systems can be unique, infinite, or nonexistent, depending on how the equations interact.

Elimination Method

This is a technique used to solve systems of linear equations by strategically adding or subtracting equations to remove one variable at a time, reducing the system to a single equation with one variable. Eliminating variables makes it easier to solve for the remaining variables and can simplify the process of identifying the overall solution to the system.

Consistency and Inconsistency of Equations

This concept refers to whether a set of equations has at least one solution (consistent) or none at all (inconsistent). A consistent system will have overlapping solution sets for all equations, whereas an inconsistent system, due to contradictory conditions, will have no single set of variable values that satisfies every equation simultaneously.

Transcript

00:01 In order to solve this system of equations, i will be using the elimination method.

00:06 In order to use the elimination method, i need to have opposite coefficients.

00:11 So my first step will be to multiply the top equation by 2.

00:19 Now when i do that, i have to make sure i multiply each term in my equation.

00:25 So let's rewrite this off to the side.

00:28 So i have 1 .5x times 2, which is just x plus 1 3rd y times 2 is 2 thirds y and 3 times 2 is 6.

00:46 Now in doing that, i chose to have opposite coefficients on my y variable.

00:52 So my second equation stayed the same.

00:57 And you'll notice i have a positive 2 thirds y and a negative 2 thirds y and a negative 2 thirds y.

01:03 So my next step is to add my equations together, so those will cancel out.

01:10 Add all your other like terms.

01:12 So one -fourth x plus x, you could think of this as 4 over 4, hence 1.

01:20 That'll be 5 over 4, x.

01:25 And then 6 plus a negative 1 is 5.

01:31 My next step will be to multiply both sides by the reciprocal.

01:36 The reciprocal means to flip the fraction, so i'm going to multiply by 4 over 5, again to both sides.

01:45 In doing that, 4 over 4 will cancel out, and 5 over 5 will cancel out.

01:50 So i'm left with just x...

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