Video: Finding the Coefficients of a System of Equations given Its Solution Set

Given that the solution set of the simultaneous equations βˆ’2π‘₯ + 9𝑦 + 2𝑧 = π‘Ž, βˆ’4π‘₯ + 9𝑦 βˆ’ 3𝑧 = 𝑏, 4π‘₯ βˆ’ 3𝑦 + 8𝑧 = 𝑐 is {(βˆ’6, 7, 8)}, find the values of π‘Ž, 𝑏, and 𝑐.

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Video Transcript

Given that the solution set of the simultaneous equations negative two π‘₯ plus nine 𝑦 plus two 𝑧 equals π‘Ž, negative four π‘₯ plus nine 𝑦 minus three 𝑧 equals 𝑏, four π‘₯ minus three 𝑦 plus eight 𝑧 equals 𝑐 is negative six, seven, eight, find the values of π‘Ž, 𝑏, and 𝑐.

So now, looking at this problem, the key to solving is actually the solution set that we’ve been given, which is negative six, seven, and eight because what this tells us is that π‘₯ is equal to negative six, 𝑦 is equal to seven, and 𝑧 is equal to eight, which is really useful cause we can actually now substitute these values in to actually determine what π‘Ž, 𝑏, and 𝑐 are.

So what I’m gonna do is I’m actually gonna start with equation one. So therefore, what we get is negative two multiplied by negative six because our π‘₯ was negative six plus nine multiplied by seven and that’s cause our 𝑦 was seven and then plus two multiplied by eight because our 𝑧 was eight is equal to π‘Ž. So we get 12 plus 63 plus 16 is equal to π‘Ž. So therefore, we get π‘Ž is equal to 81.

Okay, great, so we found the first value which is π‘Ž. So now, what we can do is actually move on to equation two to find 𝑏. So for equation two, we’ve got negative four π‘₯ plus nine 𝑦 minus three 𝑧 is equal to 𝑏. So therefore, what we get is negative four multiplied by negative six plus nine multiplied by seven minus three multiplied by eight is equal to 𝑏, which is gonna give us 24 plus 63 minus 24 equals 𝑏. And we got 24 as the first number because we had negative four multiplied by negative six. And a negative multiplied by a negative gives us positive.

So be careful of this. Be careful of your negative numbers because this is where the most common mistakes are often made So as you can see, we’ve got 24 minus 24. So these will actually cancel out. So what we’re left with is 𝑏 is equal to 63. Okay, great, so we found π‘Ž, we found 𝑏. So now, let’s move on to equation three and find 𝑐.

So for equation three, we’ve got four π‘₯ minus three 𝑦 plus eight 𝑧 is equal to 𝑐. So therefore, if we actually substitute our values of π‘₯, 𝑦, and 𝑧 in, we get four multiplied by negative six minus three multiplied by seven plus eight multiplied by eight is equal to 𝑐, which is gonna give us negative 24. So we’ve got four multiplied by negative six is negative 21 minus 21 plus 64 is all equal to 𝑐. So therefore, when we calculate this, we get 𝑐 is equal to 19.

So therefore, we can say that given that the solution set of the simultaneous equations negative two π‘₯ plus nine 𝑦 plus two 𝑧 equals π‘Ž, negative four π‘₯ plus nine 𝑦 minus three 𝑧 equals 𝑏, and four π‘₯ minus three 𝑦 plus eight 𝑧 equals 𝑐 is negative six, seven, eight, then the values of π‘Ž, 𝑏, and 𝑐 are 81, 63, and 19, respectively.

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