Equation Solving

Abstract

After reading this chapter you know:

• what equations are and the different types of equations,

• how to solve linear, quadratic and rational equations,

• how to solve a system of linear equations,

• what logarithmic and exponential equations are and how they can be solved,

• what inequations are and

• how to visualize equations and solve them graphically.

Appendices

Symbols Used in This Chapter (in Order of Their Appearance)

=	equal to
→	implies
≠	not equal to
ℂ	complex numbers
log	logarithm
ln	natural logarithm (base e)
i	complex unity
<	less than
>	greater than
≤	less than or equal to
≥	greater than or equal to
≠	not equal to
$x\in \left(-\frac{1}{2},\frac{1}{2}\right)$	$-\frac{1}{2}<x<\frac{1}{2}$
$x\in \left[-\frac{1}{2}\right.,\left.\frac{1}{2}\right)$	$-\frac{1}{2}\le x<\frac{1}{2}$
$x\in \left[-\frac{1}{2},\left.\frac{1}{2}\right]\right.$	$-\frac{1}{2}\le x\le \frac{1}{2}$
∞	infinity
∪	unification

Overview of Equations for Easy Reference

General form of linear equation

Any linear equation with one unknown x can be written as ax = b where a and b are constants.

Arithmetic rules useful for solving linear equations

If a = b and c ∈ ℂ then

$$ {\displaystyle \begin{array}{c}a+c=b+c\\ {} ac= bc\end{array}} $$

If a = b, c ∈ ℂ and c ≠ 0 then

$$ \frac{a}{c}=\frac{b}{c} $$

If F is any function and a = b then F(a) = F(b)

$$ F(a)=F(b) $$

Cramer’s rule for a system of 2 linear equations

The system of two equations with two unknowns x and y

$$ ax+ by=c $$

$$ dx+ ey=f $$

has the solution

$$ x=\left( ce- bf\right)/\left( ae- bd\right) $$

$$ y=\left( af- cd\right)/\left( ae- bd\right) $$

when

$$ ae- bd\ne 0. $$

Quadratic equation rule

The solution of

$$ {ax}^2+ bx+c=0 $$

is given by

$$ {x}_{1,2}=\frac{-b\pm \sqrt{b^2-4 ac}}{2a} $$

Factor multiplication rule

General rule:

$$ \left(x+y\right)\left(x+z\right)={x}^2+\left(y+z\right)x+ yz $$

Special cases:

$$ {\left(x+y\right)}^2={x}^2+2 xy+{y}^2 $$

$$ {\left(x-y\right)}^2={x}^2-2 xy+{y}^2 $$

$$ {x}^2-{y}^2=\left(x-y\right)\left(x+y\right) $$

Rules for changing the direction of the inequality symbol

Inequality symbol	Inequality symbol after multiplication by −1
<	>
>	<
≤	≥
≥	≤

Answers to Exercises

• 2.1. Suppose each of the granddaughters inherits x coins. Then we can write $3x+\frac{1}{2}\cdot 60=60\to 3x=60-30\to 3x=30\to x=10$. Thus each granddaughter will inherit ten rare coins.

• 2.2.

  • a. $x=-8$

  • b. $x=-23$

  • c. $3x-9=33\to 3x=33+9\to x=\frac{42}{3}\to x=14$

  • d. $x=\frac{65}{5}=13$

  • e. $4x-6=6x\kern0.75em \to \kern0.75em \left(4-6\right)x-6=0\kern0.75em \to -2x=6\kern0.75em \to \kern0.75em x=-3$

  • f. $8x-1=23-4x\kern0.75em \to \kern0.75em 12x=24\kern0.75em \to x=2$

• 2.3.

  • a) $x=0$

  • b) $x=\frac{-9}{13}$

• 2.4.

  • a) $x=4;y=0$

  • b) $x=1;y=-\frac{3}{2}$

  • c) $x=\frac{3}{2};y=-1$

  • d) $x=3;y=0$

• 2.5.

  • a) $x=2;y=-3$

  • b) $x=1;y=2$

  • c) $x=10;y=3$

  • d) $x=2;y=-1;z=1$

• 2.6.

  • a) ${x}_1=-2\frac{1}{2}\ \mathrm{and}\ {x}_2=-1$

  • b) ${x}_1=-3\ \mathrm{and}\ {x}_2=3$

  • c) ${x}_1=\frac{5-3\sqrt{5}}{2}\ \mathrm{and}\ {x}_2=\frac{5+3\sqrt{5}}{2}$

  • d) ${x}_1=\frac{5-\sqrt{37}}{2}\ \mathrm{and}\ {x}_2=\frac{5+\sqrt{37}}{2}$

  • e) ${x}_1=3-\frac{1}{2}\sqrt{42}\ \mathrm{and}\ {x}_2=3+\frac{1}{2}\sqrt{42}$

  • f) ${x}_1=-3-\frac{3}{2}\sqrt{3}\ \mathrm{and}\ {x}_2=-3+\frac{3}{2}\sqrt{3}$

• 2.7.

  • a) (x − 5)(x − 2) = 0 thus x ₁ = 2 and x ₂ = 5

  • b) (x − 4)(x − 1) = 0 thus x ₁ = 1 and x ₂ = 4

  • c) (x + 4)(x − 2) = 0 thus x ₁ =  − 4 and x ₂ = 2

  • d) We can rewrite this equation to 2x(x − 5)(x − 2) = 0 thus x ₁ = 0, x ₂ = 2 and x ₃ = 4

• 2.8.

  • a) Using that 9 = 3², we have to solve 6x = 2 and thus $x=\frac{1}{3}$.

  • b) We can rewrite 6^2x + 1 = 36 = 6² so that we have to solve 2x + 1 = 2 and thus $x=\frac{1}{2}$.

  • c) Substitute y = e ^x, so that y ² = e ^2x, resulting in the quadratic equation 10y ²− 30y + 15 = 0, which has solutions ${y}_1=\frac{3}{2}+\frac{1}{2}\sqrt{3}\ \mathrm{and}\ {y}_2=\frac{3}{2}-\frac{1}{2}\sqrt{3}$. Then ${x}_1=\ln \left(\frac{3}{2}+\frac{1}{2}\sqrt{3}\right)\mathrm{and}\ {x}_2=\ln \left(\frac{3}{2}-\frac{1}{2}\sqrt{3}\right)$.

• 2.9.

  • a) First we substitute y = log₅ x. Then the equation can be rewritten as y − y ² = 0, which has two solutions y ₁ = 0 and y ₂ = 1. Hence log₅ x ₁ = 0 and log₅ x ₂ = 1 and thus x ₁ = 1 and x ₂ = 5

  • b) We can rewrite this equation using the rules for logarithms to ${\mathrm{log}}_5\frac{{\left(3x-1\right)}^2}{12x+1}={\mathrm{log}}_51$, so that we have to solve the equation $\frac{{\left(3x-1\right)}^2}{12x+1}=1$, which has solutions x ₁ = 0, x ₂ = 2.

• 2.10.

  • a) $x\in \left(-12,\right.\left.\infty \right)$

  • b) $x\in \left(\frac{5}{12},\right.\left.\infty \right)$

  • c) $x\in \left[\frac{4}{7},\right.\left.\frac{6}{7}\right)$

• 2.11.

  • a) We first solve the equation 2x ² + 7x + 5 = 0 which has two solutions ${x}_1=\frac{-5}{2}$ and x ₂  =  − 1. Because the coefficient of x ² is larger than zero, a > 0, the parabola has a minimum and is concave up. We are looking for those solutions where the curve is ≥0. Thus

    $$ x\in \left(-\infty, \right.\left.\frac{-5}{2}\right]\cup \left[-1,\right.\left.\infty \right) $$

  • b) In a similar approach as for Exercise 2.11a we find that:

    $$ x\in \left(-3,\right.\left.3\right) $$

  • c) We first rewrite the inequation to x ²− 5x − 5 > 0. Its related equation has two solutions ${x}_{1,2}=\frac{5\pm \sqrt{45}}{2}$. Thus

    $$ x\in \Big(-\infty, \left.\frac{5-3\sqrt{5}}{2}\right)\cup \left(\frac{5+3\sqrt{5}}{2},\infty \right) $$

Glossary

Analgesic

medication to relief pain; painkiller

Algebraic

using an approach in which only mathematical symbols and arithmetic operations are used

Antipsychotic

medication used to treat psychosis

Arithmetic

operations between numbers, such as addition, subtraction, multiplication and division

Concave

hollow inward

Determinant

a scalar calculated from a matrix; can be seen as a scaling factor when calculating the inverse of a matrix (see also Sect. 5.3.1)

Elimination

eliminating an unknown by expressing it in terms of other unknowns

Equation

a mathematical expression that states that two quantities are equal

Equivalent dose

dose which would offer an equal effect between different medications

Function

a mathematical relation, like a recipe, describing how to get from an input to an output

Independent

here: equations that cannot be transformed into each other by multiplication

Least common denominator

the least number that is a multiple of all denominators

Linear

a function or mathematical relationship that can be represented by a straight line in 2D and a plane in 3D; can be thought of as ‘straight’

Nonlinear

not linear

Numerically (solving)

to find an approximate answer to a mathematical problem using computer algorithms

Polynomial

an expression consisting of a sum of products of different variables raised to different non-negative integer powers

Psychosis

a mental condition that can have many different symptoms including hallucinations

Rational equation

equation that has a rational expression on one or both sides in which the unknown variable is in one or more of the denominators

Root

solution of a polynomial equation

Substitution

replacing a symbol or variable by another mathematical expression

Transcendental

a number that is not the root of a polynomial with integer coefficients; most well-known are e and π

Union

the union of two sets is the set that contains all elements in both sets

Unique

here: a single solution to an equation

Unknown

variable in an equation for which the equation has to be solved; an equation can have multiple unknowns

Variable

alphabetic character representing a number

Vertex

peak of a parabola

Y-intercept

intercept of a curve with the y-axis