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Find all solutions to the simultaneous equations 𝑦 plus 𝑥 equals seven and 2𝑥 squared plus 𝑥 plus three 𝑦 equals 21.
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To solve simultaneous equations, there’s a couple of different ways.
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We can solve by graphing them and wherever they intersect will be their solutions, we could use elimination, we could use substitution.
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Since one of our equations is linear, the 𝑦 plus 𝑥 equals seven, it’s pretty small and it would be easy to solve for 𝑥 or 𝑦, just to isolate it.
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So we could use substitution because once we would have isolated one of those variables, we can plug it into the other one and solve.
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So let’s go ahead and take this equation and subtract 𝑥 from both sides.
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So we get that 𝑦 is equal to negative 𝑥 plus seven.
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And we can take that value for 𝑦 and plug that in for 𝑦 into our other equation, and that’s why it’s called substitution.
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Now, that we’ve substituted this in, this entire equation is in terms of 𝑥; so that’s good, so that means we can solve for 𝑥 now.
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Let’s go ahead and distribute the three.
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Three times negative 𝑥 is negative three 𝑥 and three times seven is 21.
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And now, we bring down the rest of the equation.
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So now, let’s combine like terms.
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We can put the 𝑥s together, which would be negative two 𝑥, bring down the two 𝑥 squared, and bring the 21 on the right side of the equation over to the left side.
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And they simply cancel, so we are left with two 𝑥 squared minus two 𝑥 equals zero.
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So now to solve for 𝑥, we can factor.
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So first let’s find the greatest common factor, something we can take out of both terms, and that would be two 𝑥.
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So if we would take out two 𝑥 from two 𝑥 squared, we would have 𝑥 left.
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And then if we took two 𝑥 out of negative two 𝑥, we would have a negative one left.
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Now to solve, we said each factor equals zero.
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So we said two 𝑥 equal to zero and 𝑥 minus one equal to zero.
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So we need to divide both sides by two for our first equation.
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And we get 𝑥 equals zero.
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And on our other equation, we need to add one to both sides.
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So we get one.
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So this means 𝑥 can equal zero and it can also equal one.
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So what would be the 𝑦-values?
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So we need to take zero and one and plug it into one of the original equations, so it wouldn’t matter which one.
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Let’s go ahead and plug it into the linear equation.
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Now, these two equations are the exact same thing.
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So our second equation that we kinda have moved 𝑥 over to the right-hand side, it’s equal to the original.
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So let’s go ahead and use that second one because 𝑦 is already by itself, and that’s what we’re trying to solve for.
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So when we plug in 𝑥 equals zero, we get seven for 𝑦.
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So zero, seven is one of the solutions.
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And when we plug in one, we get six.
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So one, six is another solution.
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Therefore, all of the solutions to simultaneous equations would be zero, seven and one, six.