To identify non-proportional relationships from the given equations, we should look for equations that do not pass through the origin (0,0) or have a constant ratio between \(y\) and \(x\). Proportional relationships are of the form \(y = kx\) where \(k\) is a constant, and these relationships will always pass through the origin. Here are the given equations evaluated:
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\(y = 10x + 10\): This is a linear equation with a y-intercept of 10, meaning it does not pass through the origin. (Non-proportional)
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\(y = 4.08x\): This is in the form \(y = kx\) where \(k = 4.08\). This passes through the origin. (Proportional)
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\(y = -15x\): This is also in the form \(y = kx\) where \(k = -15\). This passes through the origin. (Proportional)
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\(y = 5/x - 9\): This is a rational function that does not represent a constant ratio between \(y\) and \(x\). It does not pass through the origin. (Non-proportional)
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\(y = \frac{2}{3}x\): This is also in the form \(y = kx\) where \(k = \frac{2}{3}\). This passes through the origin. (Proportional)
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\(y = x^2\): This is a quadratic function and does not represent a constant ratio. It does not pass through the origin (since if \(x = 0\), \(y = 0^2 = 0\), but the proportionality does not hold for other values); hence, it is non-proportional. (Non-proportional)
Based on the analysis, the three equations that represent non-proportional relationships are:
- \(y = 10x + 10\)
- \(y = \frac{5}{x} - 9\)
- \(y = x^2\)