Question - Finding Equations and Simplifications
Solution:
a. To find the equation of a line that passes through points A(3, 4) and B(0, -2), use the slope formula:
m = (y2 - y1) / (x2 - x1) = (-2 - 4) / (0 - 3) = -6 / -3 = 2.
Using point-slope form: y - y1 = m(x - x1), we get:
y - 4 = 2(x - 3).
Rearranging gives: y = 2x - 6 + 4 = 2x - 2.
b. The sum of 7244 + 809871 + 19 is: 7244 + 809871 + 19 = 817134.
c. For simplification of 16^x * 2^4 * y^2z:
16 = 2^4, so 16^x = (2^4)^x = 2^(4x).
Thus, 16^x * 2^4 * y^2z = 2^(4x + 4) * y^2 * z.
d. The truth set of 8 - m = 6 is:
8 - m = 6 ⇒ m = 2.
On the number line, the truth set is {2}.
Tutor Solution:
To find the equation of a line passing through two points, we can use the slope-intercept form of a line, which is given by:
y = mx + b
where \(m\) is the slope and \(b\) is the y-intercept.
First, we need to calculate the slope \(m\).
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Using our points \(A(3, 4)\) and \(B(0, -2)\), we set:
\(x_1 = 3, y_1 = 4\), \(x_2 = 0, y_2 = -2\).Substituting the values into the slope formula:
\[ m = \frac{-2 - 4}{0 - 3} = \frac{-6}{-3} = 2 \]Now, using the slope-intercept formula:
Substituting \(m\) into \(y = mx + b\) and using one of the points to solve for \(b\). Let's use point \(A(3, 4)\):
\[ 4 = 2(3) + b \] \[ 4 = 6 + b \implies b = 4 - 6 = -2 \]Hence, the equation of the line is:
\[ y = 2x - 2 \] --- ### Question 3b **Find the sum of 7244 + 809871 + 1 leaving your answer in one decimal place.**To find the sum, we simply add the numbers together.
\[ 7244 + 809871 + 1 = 817116 \]The result is an integer and does not require decimal places, but to leave it in one decimal place, we can write:
\[ 817116.0 \] --- ### Question 3c **Simplify \(16x^2 + 4y^2z\)**.To simplify this expression, we can factor out common factors if possible.
In this case, we can factor out 4:
\[ 16x^2 + 4y^2z = 4(4x^2 + y^2z) \] --- ### Question 3d **Find the truth set of \(8 - m = 6\) and represent your answer on the number line.**We need to solve the equation for \(m\):
\[ 8 - m = 6 \]Rearranging gives:
\[ -m = 6 - 8 \] \[ -m = -2 \implies m = 2 \]The truth set is simply the value \(m = 2\). On the number line, we would represent this with a single point at 2.
--- ### Question 4a **Expand and simplify \(2(5x - 10) + 4(3x - 20) + 5\)**.Distribute each term in the equation:
\[ 2(5x - 10) + 4(3x - 20) + 5 = 10x - 20 + 12x - 80 + 5 \]Combine like terms:
\[ (10x + 12x) + (-20 - 80 + 5) = 22x - 95 \] --- ### Question 4b **Factorize \(225a^2b^2 - 5ab^2\)**.First, we can factor out the common term \(5ab^2\):
\[ 225a^2b^2 - 5ab^2 = 5ab^2(45a - 1) \] --- ### Question 4c **Find the magnitude/length of \(X(1, 3)\) and \(B(2, -7)\)**.The distance formula between two points \(X(x_1, y_1)\) and \(B(x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Plugging in our points \(X(1, 3)\) and \(B(2, -7)\):
\[ d = \sqrt{(2 - 1)^2 + (-7 - 3)^2} = \sqrt{1^2 + (-10)^2} = \sqrt{1 + 100} = \sqrt{101} \] --- ### Question 4d **The angles of a pentagon are \(x^0, (x + 20)^0, (x + 25)^0, 2x^0\) and \((2x + 25)^0\). Find \(x\).**The sum of the angles in a pentagon is \(540^\circ\). Therefore, we have:
\[ x + (x + 20) + (x + 25) + 2x + (2x + 25) = 540 \]Combining like terms:
\[ 5x + 90 = 540 \]Subtracting 90 from both sides gives:
\[ 5x = 450 \implies x = 90 \] --- ### Question 5a **Express 84 as a product of its prime factors.**The prime factorization of 84 is:
\[ 84 = 2^2 \times 3 \times 7 \] --- ### Question 5b **List the members of each of the sets \(M = \{factors \, of \, 63\}\) and \(N = \{factors \, of \, 72\}\)**.To find the factors of 63:
The factors of 63 are \(1, 3, 7, 9, 21, 63\).
For 72, the factors are \(1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72\).
In conclusion:
M = \{1, 3, 7, 9, 21, 63\}
N = \{1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72\}
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