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3.5 Solving Systems of Linear Equations
https://math.libretexts.org/Courses/Siena_College/Preparation_for_College_Mathematics/Chapter_3%3A_Equations_and_Inequalities/3.5_Solving_Systems_of_Linear_Equations
I know that in order to be perfectly balanced, the sum of the moments (turning effect of the weight of each block on a lever, aka the weight of the block multiplied by the distance from the fulcrum) m...I know that in order to be perfectly balanced, the sum of the moments (turning effect of the weight of each block on a lever, aka the weight of the block multiplied by the distance from the fulcrum) must balance out on either side of the plank. A system of equations is a set of equations that have the exact same variables, for which a solution is a set of values chosen for each variable that will satisfy all of the equations simultaneously.
1.2E Exercises
https://math.libretexts.org/Courses/Siena_College/Preparation_for_College_Mathematics/01%3AComputation/1.2_Working_With_Fractions/1.2E_Exercises
Remember I said you can only split up a fraction like $\frac{a+b}{c} = \frac{a}{c}+\frac{b}{c}$ because the addition is in the numerator? The left hand side is \( \frac{a}{d} - \frac{b+c}{d} = \fr...Remember I said you can only split up a fraction like $\frac{a+b}{c} = \frac{a}{c}+\frac{b}{c}$ because the addition is in the numerator? The left hand side is $\frac{a}{d} - \frac{b+c}{d} = \frac{2}{4} - \frac{1+3}{4} = \frac{1}{2} - \frac{4}{4} = -\frac{1}{2}$ , but the hypothesized right hand side would be $\frac{2 - 1 + 3}{4} = \frac{4}{4} = 1.$ Not the same! $\frac{a}{d} - \frac{b+c}{d} =\frac{a}{d} - \frac{(b+c)}{d} \notag$
Licensing
https://math.libretexts.org/Courses/Siena_College/Preparation_for_College_Mathematics/00%3A_Front_Matter/04%3A_Licensing
A detailed breakdown of this resource's licensing can be found in Back Matter/Detailed Licensing.
Table of Contents
https://math.libretexts.org/Courses/Siena_College/Preparation_for_College_Mathematics/01%3AComputation/00%3A_Front_Matter/03%3A_Table_of_Contents
3.7 Chapter 3 Study Guide
https://math.libretexts.org/Courses/Siena_College/Preparation_for_College_Mathematics/Chapter_3%3A_Equations_and_Inequalities/3.7_Chapter_3_Study_Guide
All of the points $( x,y)$ lying on a straight line will satisfy the equation of the line, meaning they can be plugged in and the equation will be true, and meaning that the line is made up of all t...All of the points $( x,y)$ lying on a straight line will satisfy the equation of the line, meaning they can be plugged in and the equation will be true, and meaning that the line is made up of all the points $(x,y)$ that do satisfy the equation. To see if an equation with $x^2$ and $y^2$ terms is the equation of a circle, complete the square on $x$ terms and $y$ terms separately to see if it can be made to match the standard form.
6.5 Chapter 6 Study Guide
https://math.libretexts.org/Courses/Siena_College/Preparation_for_College_Mathematics/Chapter_6%3A_Intro_to_More_Functions/6.5_Chapter_6_Study_Guide
limit of $f$ at $x = a$ : the $y$ -value that the function's graph is approaching as $x$ approaches $a$ from both the left and the right. When looking at a graph of $y = f(x)$ , the limit o...limit of $f$ at $x = a$ : the $y$ -value that the function's graph is approaching as $x$ approaches $a$ from both the left and the right. When looking at a graph of $y = f(x)$ , the limit of $f$ at $x = a$ is the $y$ -value that the function appears to be heading toward as you move towards a particular $x$ -value of interest, $a$ , from both the left and the right.
00: Front Matter
https://math.libretexts.org/Courses/Siena_College/Preparation_for_College_Mathematics/01%3AComputation/00%3A_Front_Matter
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3.3 Lines and Circles
https://math.libretexts.org/Courses/Siena_College/Preparation_for_College_Mathematics/Chapter_3%3A_Equations_and_Inequalities/3.3_Lines_and_Circles
Points on the plane are defined by a coordinate pair of numbers, $(x, y)$ , where the first coordinate is always the $x$ -value and the second coordinate is always the $y$ -value. (In the olden day...Points on the plane are defined by a coordinate pair of numbers, $(x, y)$ , where the first coordinate is always the $x$ -value and the second coordinate is always the $y$ -value. (In the olden days these coordinates were called the abscissa and the ordinate, but now if you say that people will be like "Oh yeah, weren't those the two monsters they had to sail between in the Odyssey?"
3.4E Exercises
https://math.libretexts.org/Courses/Siena_College/Preparation_for_College_Mathematics/Chapter_3%3A_Equations_and_Inequalities/3.4_Modeling_With_Equations/3.4E_Exercises
To do this, algebraically manipulate the equation to see if a variable can be written either as a constant multiple of the other ( $y= kx$ ) or as a constant multiple of the reciprocal of the other \...To do this, algebraically manipulate the equation to see if a variable can be written either as a constant multiple of the other ( $y= kx$ ) or as a constant multiple of the reciprocal of the other $\left( y = \dfrac{k}{x} \right)$ . The force $F$ needed to keep the kart from drifting is jointly proportional to the weight $w$ of the kart and the square of my speed $v$ , and inversely proportional to the radius of the curve.
2.1E Exercises
https://math.libretexts.org/Courses/Siena_College/Preparation_for_College_Mathematics/Chapter_2%3A_Algebra_Skills/2.1_Algebraic_Expressions/2.1E_Exercises
Hint: What does "Simplify" mean when a problem looks like this? $(x + 2)(x + 3)$ $(2x - 1)(x+1)$ $7xe^x + 2e^x + 7x + 2$ $(x - 1)(2x^2 + x + 1)$ $(a -b - 1)(a^2 + 2ab + b^2)$ \( a^3 -...Hint: What does "Simplify" mean when a problem looks like this? $(x + 2)(x + 3)$ $(2x - 1)(x+1)$ $7xe^x + 2e^x + 7x + 2$ $(x - 1)(2x^2 + x + 1)$ $(a -b - 1)(a^2 + 2ab + b^2)$ $a^3 - a^2 - 5a + 2$ $-a^2 + a^3 - 2 a b + a^2 b - b^2 - a b^2 - b^3$ $a^3 - 3 a^2 b + 3 a b^2 - b^3$ $(x+h)^2 - x^2$ $(3 + b)(3 - b) - (3+b)(3+b)$ $(3x^2)(x^2+1) + (2x)(x^3 - 2)$ $x^2 - 2x + y^2 + 2y +2$ $x + \sqrt{x} - x^2$ $-4 x + 3 x^2 + 5 x^4$ $(x+3)(x+3)$
6.1 Rational Functions
https://math.libretexts.org/Courses/Siena_College/Preparation_for_College_Mathematics/Chapter_6%3A_Intro_to_More_Functions/6.1_Rational_Functions
There are roots at $x = -1$ and $x = 0$ , which are exactly the $x$ -values that cause the numerator to be 0, which of course causes the entire fraction to be 0. This is productive if (a) the degr...There are roots at $x = -1$ and $x = 0$ , which are exactly the $x$ -values that cause the numerator to be 0, which of course causes the entire fraction to be 0. This is productive if (a) the degree of the top is larger than the degree of the bottom, or (b) the degrees are the same. $f(x) = \dfrac{ x^2 - 4 } {x^2 - 9} = \dfrac{ x^2 \textcolor{red}{-9+5} } {x^2 - 9} = \dfrac{ x^2 - 9 } {x^2 - 9} + \dfrac{ 5} {x^2 - 9} = 1 + \dfrac{ 5} {x^2 - 9} \notag$

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