3.7 Chapter 3 Study Guide

Definitions

• equation: a mathematical statement that two expressions are equal.
• solution: a set of values that, when plugged in for the variables, will make an equation's statement true.
• equivalent equations: equations that have exactly the same solutions. When this is the case, you can algebraically manipulate one equation until it matches the other.
• linear equation in \(n\) unknowns: an equation with \(n\) variables that appear only to the first power. Examples: \( 3x + 7 = 20 \), \( 4x_1 + 2x_2 - x_3 = 0 \). Non-examples: \(\sqrt{x} + y = 1 \), \(x^2 + xy = 2\).
• inverse operation: an operation that undoes another operation. Examples: addition/subtraction, multiplication/division, raising to powers/taking corresponding roots.
• opposite: the opposite of a number or expression is the same number or expression, but with the opposite sign. Examples: opposite of \(-3\) is \(3\), opposite of \(x+1\) is \( -(x+1)\).
• absolute value: the absolute value function \( |\text{inside}| \) will return \( \text{inside} \) when the inside is nonnegative, and returns the opposite, \( -(\text{inside}) \), if it's negative.
• quadratic equation: a polynomial equation where the highest power on the variable is \(2\). Standard form is \(ax^2 + bx + c = 0\).
• inequality: a mathematical statement that two expressions are related by one of the inequality signs, \(<, \leq, >, \geq\).
• simultaneous inequality: an inequality with two bookends, like \( -4 < x + 2 < 15 \).
• coordinate/Cartesian plane: the 2D plane defined by \(x\)-axis and \(y\)-axis. A point on the plane is labeled with a coordinate pair \( (x,y)\).
• origin: the point of intersection of the axes, aka coordinates \( (0,0)\).
• intercepts: an \(x\)-intercept is a point where a graph crosses the \(x\)-axis; a \(y\)-intercept is a point where a graph crosses the \(y\)-axis.
• slope: describes the steepness of a line or the rate of vertical change relative to the horizontal change. Denoted \(m = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1} \). If slope is positive, the line runs uphill. If negative, the line runs downhill.
• slope-intercept form: an equation of a line in the form \( y = mx + b\), where \(m\) gives the slope, and the \(y\)-intercept is \( (0,b)\).
• point-slope form: an equation of a line in the form \(y - y_0 = m(x - x_0 )\), where \(m\) is the slope and the line passes through point \( (x_0,y_0) \).
• standard equation of a circle: an equation of the form \((x-h)^2 + (y-k)^2 = r^2 \), where \( (h,k) \) is the center of the circle and \(r\) is the radius.
• mathematical model: a way of describing a real-world situation with an equation or formula. Variables are used to represent quantities.
• direct variation: if \(y = kx\), where \(k\) is some constant of proportionality, then \(y\) varies directly as or is directly proportional to \(x\).
• inverse variation: if \(y = \dfrac{k}{x}\), where \(k\) is some constant of proportionality, then \(y\) varies inversely as or is inversely proportional to \(x\).
• system of linear equations: a set of linear equations using the same exact variables, for which we look for a solution, a set of values for the variables that will satisfy all of the equations simultaneously.
• substitution: the method of solving a system that involves solving an equation for a single variable, then substituting that expression into the other equation(s).
• elimination: the method that involves multiplying equation(s) by constants to achieve the cancelling out of a variable entirely when the equations are added.
• inconsistent: if a system has no solutions, it is called inconsistent.
• \( \mathbb{R}^2\) (\( \mathbb{R}^3\)): the set of all pairs (or triples, respectively) of real numbers, which are written as...
• vectors: written like \( \begin{bmatrix} a \\ b\end{bmatrix} \) or \( \begin{bmatrix} a \\ b \\ c \end{bmatrix} \). These can be thought of as arrows pointing from the origin to a point with coordinates \( (a,b)\) (in 2D space) or \( (a,b,c) \) (in 3D space).
• \(n \times m\) matrix: an array of entries with \(n\) rows and \(m\) columns. A \(2 \times 2 \) matrix looks like \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).
• \(n \times n\) identity matrix: a square matrix with 1s on the diagonal entries and 0s elsewhere, such as \( I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \). Multiplying a vector by an identity matrix will not change the vector: \(Iv = v\).
• inverse of a matrix: the inverse of a square matrix \(A\) is another square matrix \(A^{-1}\) such that \( A^{-1} (Av) = v \), or in other words, \(A A^{-1} = I\).

Solving a Quadratic Equation

To solve a quadratic equation in the standard form \(ax^2 + bx + c = 0\), there are three main techniques. Ranked from most convenient to least convenient, on average, they are:

1. Basic factoring when \(a = 1\). This is the "find two numbers that multiply to \(c\) and add to \(b\)" method. If \(a \neq 1\), try the \(ac\) method to factor by grouping. After factoring, you have an expression containing factors in the form \((x - c)\) multiplied together, with zero on the other side. Each such factor has the potential to wipe out the entire expression were it to equal 0, and \(x = c\) would cause that! Then each factor gives a solution to the equation.
2. Completing the square on the first two terms, while moving the constant term \(c\) to the other side, gives an equation of the form \( (x+m)^2 = n\). Then by taking a square root of both sides and subtracting the \(m\), we can isolate \(x\).
3. Solutions can always be found using the quadratic formula:

\[ x = \dfrac{ -b \pm \sqrt{b^2 - 4ac} }{2a} = \dfrac{ -b + \sqrt{b^2 - 4ac} }{2a}, \dfrac{ -b - \sqrt{b^2 - 4ac} }{2a} \notag \]

Possible Outcomes When Solving Quadratic Equations

• Distinct real roots: either something like "\( (x- 3)(x-2) = 0 \quad \implies \quad x = 2, 3 \)," or a result like \(x = 2 \pm \sqrt{3} \).
• Repeated real root: two copies of the same number, coming from something like "\( (x+1)^2 = 0\)."
• No real roots, but a complex conjugate pair: when the discriminant \(b^2 - 4ac\) turns out to be negative, these will look like \(x = a \pm bi\).

Caution

Get in the habit of checking your answers by plugging them back into the original equation or inequality. Sometimes when solving radical equations or absolute value equations, extraneous invalid solutions appear that must be thrown out!

Algebra Rules for Inequalities

• You can add or subtract the same thing on both sides of an inequality, and it won't change: \(A < B \iff A + C < B + C \).
• You can multiply or divide something positive on both sides of an inequality, and it won't change: if \(C \geq 0\), \( A < B \iff AC < BC \).
• ALERT!!! If you multiply or divide both sides by a negative number, you must flip the direction of the inequality sign: if \(C < 0\), \(A<B \iff AC > BC\) or \( \frac{A}{C} > \frac{B}{C} \).

Inequalities Involving Absolute Value

• For \( < \) and \(\leq \), split into cases using AND: \( |\text{inside}| < a \quad \implies \quad (\text{inside})<a \) AND \( -(\text{inside}) < a \).
• For \( >\) and \( \geq\), split into cases using OR: \( |\text{inside}| > a \quad \implies \quad (\text{inside})> a \) OR \( -(\text{inside}) > a \)

Caution

AVOID THIS COMMON MISTAKE: \( |3-x| \textcolor{red}{\neq} 3+x \) !!!!!! It splits into two situations: \( (3-x)\) if \( (3-x) \geq 0\), and \( -(3-x) = x - 3 \) if \( (3-x) < 0\).

Fun Facts About Lines

• An \(x\)-intercept is a point where the line crosses the \(x\)-axis, and it will always have a \(y\)-coordinate of 0.
• A \(y\)-intercept is a point where the line cross the \(y\)-axis, and it will always have an \(x\)-coordinate of 0.
• The slope \(m\) of a line can be calculated using two distinct points on the line, \( (x_1,y_1)\) and \( (x_2,y_2)\), by computing rise over run:
  \[ m = \dfrac{ \text{rise}}{\text{run}} = \dfrac{ \text{change in y}}{\text{change in x}} = \dfrac{y_2 - y_1}{x_2-x_1} \notag \]
• Uphill / Increasing lines have positive slopes. The bigger the number, the steeper the line.
• Downhill / Decreasing lines have negative slopes. Again, whole numbers more negative than \( -1\) make steeper lines and little fractional slopes like \(-\frac{1}{2} \) make less steep lines.
• If the slope of a line is zero, it's actually a flat horizontal line.
• The slope of a vertical line is undefined.
• 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.
• The equation of a line with slope \(m\) and \(y\)-intercept \( (0,b)\) is \( y = mx + b \).
• The equation of a line with slope \(m\) passing through the point \( (x_0,y_0)\) is \( y - y_0 = m ( x - x_0 ) \).
• Parallel lines have the same slope.
• If two lines are perpendicular to each other then their slopes are negative reciprocals. (The negative reciprocal of a number \( \frac{a}{b} \) is \(- \frac{b}{a} \).)
• To find where two lines \(y = m_1 x + b_1 \) and \(y = m_2 x + b_2\) intersect, set \(m_1 x + b_1 = m_2 x + b_2 \) and solve for \(x\). Then to find the corresponding \(y\)-coordinate, plug that \(x\) solution into either of the original equations.
• To sketch the graph of a line:
  • If you know one point and the slope \(m = \frac{a}{b}\) is positive, plot the point and then find another point just by tracking \(b\) units to the right, and then \(a\) units up. Drop a point in right there, connect them with a straight line, and you're done.
  • If the slope \(m = \frac{a}{b}\) is negative, then track \(b\) units to the right, and then \(a\) units down instead.
  • If a slope-intercept form has no \(b\) term, like this \(y = 3x\), you automatically know that line passes through the origin since its \(y\)-intercept is \( (0,0)\).
  • The equation of a horizontal line passing through \(b\) on the \(y\)-axis is \(y = b\).
  • The equation of a vertical line passing through \(a\) on the \(x\)-axis is \(x = a\).

Fun Facts About Circles

• The equation of a circle centered at the point \( (h,k) \) with radius \(r\) is \( (x-h)^2 + (y-k)^2 = r^2 \).
• The radius is the uniform distance from the center to any point on the circle.
• The diameter is the distance from one side of the circle across to the other, passing through the center, so it's double the radius. That is, diameter \( d = 2r\).
• The area of a circle with radius \(r\) is \(\pi r^2\).
• The circumference of a circle is the distance all the way around it, as in how many steps your Apple watch would say you covered if you walked the whole perimeter, and it's calculated \(2\pi r\).
• To sketch the graph of a circle, identify and plot the center, and then draw four cardinal points at the appropriate distance away for the radius. Connect the dots with curves as best you can to create a circle shape.
• 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.

Variation

• Direct variation: if \(y = kx\), where \(k\) is some constant of proportionality, then \(y\) varies directly as or is directly proportional to \(x\).
• Inverse variation: if \(y = \frac{k}{x}\), where \(k\) is some constant of proportionality, then \(y\) varies inversely as or is inversely proportional to \(x\).
• If \(x,y,\) and \(z\) are related by the equation \(z = kxy\), we say \( z\) is proportional to the product of \(x\) and \(y\), or \(z\) is jointly proportional to \(x\) and \(y\), or \(z\) varies jointly as \(x\) and \(y\).
• Looking at an equation solved for one variable, like the above or like \(z = k \dfrac{x}{y}\), we know that \(z\) is directly proportional to any variables in the numerator, and inversely proportional to any variables in the denominator.
• You can also be proportional to expressions other than a variable alone! For example, if \(z = k(x+y) \), we say \(z\) is directly proportional to the sum of \(x\) and \(y\). Or if \(z = \dfrac{k}{x^2} \), we say that \(z\) is inversely proportional to the square of \(x\).

Solving Systems

• Substitution: Solve one equation for a particular variable, then sub in for that variable in the second equation, yielding an equation with one unknown, and solve that. Then plug that result into an original equation to find the second unknown.
• Elimination: Multiply one or both equations by whatever is needed so that, on a particular variable, the equations have opposite coefficients. Then line up all the variable terms and constant terms in columns correctly and add down the columns, yielding an equation with one unknown, and solve that. Then plug the result into an original equation to get the other unknown.
• A system of two linear equations in two unknowns can have:
  • one unique solution (the equations' lines intersect once)
  • no solutions (the equations' lines are parallel and never intersect; in this case the system is called inconsistent) This will happen if as you're solving, you end up with an equation that is a lie, like \(0 = 5\).
  • infinitely many solutions (the equations' lines are smack on top of each other, the same line really, and have all points in common) This will happen if as you're solving, you end up with all the variables killing each other off, resulting in a true equation, like \( 0 = 0\). This always happens if one equation is a constant multiple of another.

Vector Arithmetic

• Vector addition/subtraction:
  \[ \begin{bmatrix} a \\ b \end{bmatrix} + \begin{bmatrix} c \\ d \end{bmatrix} = \begin{bmatrix} a + c \\ b + d \end{bmatrix}, \quad \quad \begin{bmatrix} a \\ b \end{bmatrix} - \begin{bmatrix} c \\ d \end{bmatrix} = \begin{bmatrix} a - c \\ b - d \end{bmatrix} \notag \]
  Geometrically, you are "chaining" up the vectors nose to tail and drawing a new vector from the origin to the final arrowhead.
• Scalar multiplication: \( k \begin{bmatrix} a \\ b \\ c \end{bmatrix} = \begin{bmatrix} ka \\ kb \\ kc \end{bmatrix} \). Geometrically, this stretches or shrinks a vector's length without changing its direction. A negative scalar will cause a \(180^\circ\) flip in direction.

Matrix Arithmetic

• Matrix addition/subtraction (requires the same exact size): \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} + \begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix}a + e & f+b \\ c+g & d + h \end{bmatrix} \)
• Scalar multiplication: \( k \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix}ka & kb \\ kc & kd \end{bmatrix} \)
• \(2 \times 2\) matrix-vector multiplication of \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) and \(v = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} \) results in a new \(2 \times 1\) vector

\[ Av = \begin{bmatrix} av_1+bv_2 \\ cv_1 + dv_2 \end{bmatrix} \notag \]

Inverse of a Matrix

The inverse of a \(2 \times 2\) matrix \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by \(A^{-1} = \dfrac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\). The quantity \( ad - bc \) is called the determinant of \(A\), and must not equal zero for this to work! If a matrix's determinant turns out to be zero, it is not invertible.

Solving \(2\times 2\) Systems With Matrix Inverses

To solve a system \( \begin{cases} ax + by = v_1 \\ cx + dy = v_2 \end{cases}\),

1. Translate to a matrix equation \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix}x \\ y \end{bmatrix} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} \).

2. Find the inverse of the coefficient matrix using the formula \( \dfrac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \) as long as \(ad-bc \neq 0\). If it is 0, the system is inconsistent and has no solutions.

3. Solve for \( \begin{bmatrix}x \\ y \end{bmatrix} \) by multiplying \( \dfrac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\begin{bmatrix} v_1 \\ v_2 \end{bmatrix} \).