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Equivalent systems of equations Video transcript - [Voiceover] "Vivek and Camila's teacher gave them a system "of linear equations to solve. Which of them obtained a system that is equivalent to the teacher's system?
So the first question we should ask ourselves is what does it mean to even have an equivalent system? For the sake of this question, or for our purposes, an equivalent system is a system that has the same solution.
So if there's some X-Y pair that satisfies the teacher's system that is the solution to the teacher's system. Well Vivek's system, we're gonna call it equivalent if it has the same solution. Similarly, if Camila's system has the same solution, then we're gonna call it equivalent to the teacher's system.
So let's make some comparisons here. So first let's look at Vivek.
So his first equation is actually unchanged from the teacher's equation, is unchanged from the teacher's equation, so any solution that meets both of these equations is for sure gonna meet this top equation because it's literally the same as the top equation of the teacher, so that works out.
Also look at the second one. The second one is definitely a different equation over here. We can check that it's not just being multiplied by some number on both sides. To go from one to zero if you were multiplying, you would have to multiply one times zero and then in order to maintain the equality, you would have to do that on both sides.
But zero times this left-hand side would have been zero, you would have gotten zero equals zero, so he didn't just scale both sides by some number, looks like he did another operation. He probably looks like he's adding or subtracting something to both sides, so let's see how he could have gotten this right over here.
So he took -4x plus 5y is equal to one. And it looks like from that he was able to get - 3x plus 7y is equal to zero. So let's see what he had to do to do that. Let's see, he would have had to, to go from -4x to -3x, he would have had to add an X, so I could just write an X right over there.
To go from 5y to 7y, he would have had to add 2y. So on the left-hand side, he would have to add X plus 2y.
Notice we have an X plus 2y right over there. And on the right-hand side, he would have had to add or subtract a one, or add a negative one. Notice we see a negative one right over there.
So what he essentially did is he added the left-hand sides of these two equations to get this new left-hand side right over here, and he added the right-hand sides to get this new right-hand side. And that is a legitimate operation. This new equation that you got, this new linear equation, it's going to represent a different line than this one right over here, but the resulting system is going to have the same solution.
Why do we feel confident that the resulting system is going to have the same solution?
Well for an X-Y pair that satisfies both of these equations, that's what a solution would be, for that X-Y pair, X plus 2y is equal to negative one. So for that solution, we're adding the same thing to both sides. That's not going to change the solution of the system. In fact, that's a technique we often use to eventually find the solution of a system.
So now let's look at Camila, or Camila. So her first equation is actually the exact same equation as the teacher's second equation. Now let's see, her second equation, how does it relate possibly to the first equation? So just looking at it offhand, it looks like it might just be, it looks like she just multiplied both sides times a number.
And it looks like that number, she clearly multiplied the right-hand side times negative eight.Use substitution to determine if the given ordered pair is an element of the solution set for the system of equations.
Substitute 1 for x and 3 for y in each equation. Parametric Equations in the Graphing Calculator.
We can graph the set of parametric equations above by using a graphing calculator. First change the MODE from FUNCTION to PARAMETRIC, and enter the equations for X and Y in “Y =”.. For the WINDOW, you can put in the min and max values for \(t\), and also the min and max values for \(x\) and \(y\) if you want to.
Algebra 2 Here is a list of all of the skills students learn in Algebra 2! These skills are organized into categories, and you can move your mouse over any skill name to preview the skill. How to Balance Chemical Equations. In this Article: Article Summary Traditional Balance Algebraic Balance Community Q&A A chemical equation is a written symbolic representation of a chemical reaction.
The reactant chemical(s) are given on the left-hand . Here you can find a printable periodic table (black and white and color version), bookmark sized 'Periodic table' with basic instructions on how to balanse chemical reactions, rules and style conventions for writing SI units and quantities shortened to a single page, paper models of crystal systems or a large educational poster for your laboratory wall.
• Determine if an equation or inequality is appropriate for a given situation. • Solve mathematical and real-world problems with equations. • Represent real-world situations as inequalities.