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Eric H.
Engineering Grad from UC San Diego
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Pre-Calculus
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Question:

Simplify the logarithmic expression: $(ln(\frac{x^4}{(2x-1)^3(7x-5)^8})$)

Eric H.

To simplify this logarithmic expression, we will need a couple properties of logarithms. They are the multiplication law, the division law, and the exponent law: $(ln(x*y) = ln(x) + ln(y)$) $(ln(\frac{x}{y}) = ln(x) - ln(y)$) $(ln(x^2) = 2ln(x)$)To simplify this large expression, start with the main largest operation, which is the division of the numerator and denominator inside the expression. Using the division law, the expression becomes split into two. Use the multiplication property to split the second component, and then use the exponent law to reduce all components. $(ln(x^4) - ln((2x-1)^3(7x-5)^8))$) $(ln(x^4) - ln((2x-1)^3) - ln((7x-5)^8)$) $(4ln(x) - 3ln(2x-1) - 8ln(7x-5)$)This is the final simplified form of the expression.

ACT
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Question:

A summer camp has a total of 280 children who are either 11 or 12 years old. The sum of the children’s ages is 3,238. How many 11-year-old children are at the camp? A. 55 B. 122 C. 132 D. 158 E. 208

Eric H.

Once you see this type of problem, where you are given some information about sums or totals of a group of items, you can typically assume it will be a system of equations problem. Let's call the number of 11-year-olds x, and the number of 12-year-olds y. Let us now translate the problem now. $(Sentence\, 1: 280 = x + y$)$(Sentence\, 2: 3238 = 11x + 12y$)Now we have a system of equations that we can solve. Multiple one of the equations so that we can subtract one equations from the other and eliminate one of the variables. In this case, since we want to find the number of 11-year-olds at the camp, we should eliminate y so we can solve for x.$(3360 = 12x + 12y$)$(3238 = 11x + 12y$)Now we can simply subtract equation 2 from equation 1, leaving us:$(122 = x$)There are 122 11-year-old children at the camp.

Algebra
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Question:

Find the equation of the line that passes through the points (2 , -4) and (6 , 2).

Eric H.

Any equation of a line can be expressed in this form: $(y = mx + b$)Where: m represents the slope of the line b represents the y-intercept To find the equation of a line given two points, simply calculate m and b. m is how much the line "rises" over how much it "runs", so $(m = (y_{2}-y_{1})/(x_{2}-x_{1})$)$(=(2-(-4)){/(6-2)}$)$(= 1.5$)Once you have m, plug it into the equation y = mx + b so you have y = 1.5x + b. Now you can solve for b by plugging in one of the points given in the question. Let's use (6, 2).$(y = 1.5x + b$)$(2 = 1.5(6) + b$)$(b=-7$)With that you have now values for both m and b, so you need only plug them both in to obtain the final equation:$(y=1.5x-7$)

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