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2 years ago

What are the solutions to the system of equations? y=x2−7x+12y=−x+7

Mathematics

1 answer:

kap26 [50]2 years ago

3 0

Answer:

x = 1, y = 6

x = 5, y = 2.

Step-by-step explanation:

y=x2−7x+12

y=−x+7

Substitute for y in the first equation:

- x + 7 = x^2 - 7x + 12

x^2 - 7x + x + 12 - 7 = 0

x^2 - 6x + 5 = 0

(x - 1)(x - 5) = 0

x = 1, 5.

When x = 1, y = -1 + 7 = 6.

when x = 5, y = -5+7 = 2.

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Beth says that the graph of g(x) = + 1 is a translation of 5 units to the left and 1 unit up of f(x) = . She continues to explai

Nata [24]

Beth's description of the transformation is incorrect

<h3>Complete question</h3>

Beth says that the graph of g(x)=x-5+1 is a translation of 5 units to the left and 1 unit up of f(x) = x. She continues to explain that the point (0,0) on the square root function would be translated to the point (-5,1) on the graph of g(x). Is Beth's description of the transformation correct? Explain

<h3>How to determine the true statement?</h3>

The functions are given as:

g(x) = x - 5 + 1

f(x) = x

When the function f(x) is translated 5 units left, we have:

f(x + 5) = x + 5

When the above function is translated 1 unit up, we have:

f(x + 5) + 1 = x + 5 + 1

This means that the actual equation of g(x) should be

g(x) = x + 5 + 1

And not g(x) = x - 5 + 1

By comparison;

g(x) = x - 5 + 1 and g(x) = x + 5 + 1 are not the same

Hence, Beth's description of the transformation is incorrect

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Thompson and Thompson is a steel bolts manufacturing company. Their current steel bolts have a mean diameter of 133 millimeters,

Sophie [7]

Answer:

0.5352 = 53.52% probability that the sample mean would differ from the population mean by more than 0.8 millimeters

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation(which is the square root of the variance) $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

$\mu = 133, \sigma = \sqrt{64} = 8, n = 38, s = \frac{8}{\sqrt{38}} = 1.2978$

If a random sample of 38 steel bolts is selected, what is the probability that the sample mean would differ from the population mean by more than 0.8 millimeters?

Either it differs by 0.8 millimeters or less, or it differs by more than 0.8 millimeters. The sum of the probabilities of these events is 1.

Probability it differs by 0.8 millimeters or less.

pvalue of Z when X = 133 + 0.8 = 133.8 subtracted by the pvalue of Z when X = 133 - 0.8 = 132.2.

X = 133.8

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$