To find the solution to the system of equations \(y = -3x + 6\) and \(y = 4x - 5\), we can analyze the values Lucy provided in the table.
- The first equation gives us \(y = -3x + 6\).
- The second equation gives us \(y = 4x - 5\).
To find the point of intersection (the solution to the system), we can set the two equations equal to each other:
\[ -3x + 6 = 4x - 5 \]
Now, solve for x:
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Add \(3x\) to both sides: \[ 6 = 7x - 5 \]
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Add \(5\) to both sides: \[ 11 = 7x \]
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Divide by \(7\): \[ x = \frac{11}{7} \approx 1.5714 \]
Next, we substitute \(x = \frac{11}{7}\) back into one of the original equations to find \(y\). Using the first equation:
\[ y = -3\left(\frac{11}{7}\right) + 6 = -\frac{33}{7} + 6 = -\frac{33}{7} + \frac{42}{7} = \frac{9}{7} \approx 1.2857 \]
So the calculated intersection point is:
\[ \left(\frac{11}{7}, \frac{9}{7}\right) \approx (1.5714, 1.2857) \]
We see from Lucy's table that for \(x\) values between 1.5 and 2, the possible corresponding \(y\) values for \(x = 1.5\) give:
For \(x = 1.5\):
- From \(y = -3(1.5) + 6 = -4.5 + 6 = 1.5\)
- From \(y = 4(1.5) - 5 = 6 - 5 = 1\)
Attempting \(x = 1.6\):
- From \(y = -3(1.6) + 6 = -4.8 + 6 = 1.2\)
- From \(y = 4(1.6) - 5 = 6.4 - 5 = 1.4\)
And so on, adjusting the values through \(x = 1.7\), \(1.8\), and \(2\).
In the table that Lucy made, her results would lead her to identify at which precise \(x\) the \(y\) values get closer and where they intersect. Without more precise pairing of \(x\) and \(y\) lists from her table, it appears that \(x\) around \(1.6\), where \(y\) values closely match, is promising.
From the analysis, the best approximation from \(x = \frac{11}{7}\) and substituting would suggest the ordered pair of \( (1.5714, 1.2857) \).
Thus, the best approximation of the exact solution is:
Answer: \((1.5714, 1.2857)\) or approximately \((1.57, 1.29)\) if rounding to two decimal places.
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