Name: solving linear equations
Uploaded: 2025-11-26T05:44:31.000Z
Duration: 1 min 27 s
Channel: JoVE

Równania liniowe mogą być równaniami jedno-, dwu- lub trzyzmiennymi, w zależności od liczby niewiadomych. Równanie liniowe z jedną zmienną to równanie algebraiczne z unikalnymi stałymi i jedną zmienną.Graficznie równanie liniowe reprezentuje linię prostą na płaszczyźnie współrzędnych.Rozwiązanie równania liniowego oznacza znalezienie wartości zmiennej poprzez wyizolowanie jej tak, aby po podstawieniu wartości obie strony były równe.Na przykład firma taksówkarska pobiera stałą opłatę plus stałą stawkę za kilometr, a 10-kilometrowa przejażdżka kosztuje 35 dolarów. Celem jest określenie stawki za kilometr.Po pierwsze, nieznana stawka za kilometr jest definiowana jako zmienna. Zmienna ta jest mnożona przez liczbę kilometrów i dodawana do opłaty ryczałtowej w celu obliczenia całkowitego kosztu.Odjęcie opłaty ryczałtowej od całkowitego kosztu i podzielenie reszty przez liczbę kilometrów określa stawkę za kilometr.Wartość ta jest następnie podstawiana z powrotem do oryginalnego równania w celu zweryfikowania rozwiązania.Podobnie, obliczenie ilości zakupionego gazu na wydanego dolara ilustruje, w jaki sposób równania liniowe mają zastosowanie w rzeczywistych sytuacjach

równania liniowe

R&#243;wnania liniowe stanowi&#261; podstaw&#281; wielu zastosowa&#324; algebraicznych i rzeczywistych, charakteryzuj&#261;c si&#281; prostot&#261; i u&#380;yteczno&#347;ci&#261;. R&#243;wnanie liniowe to wyra&#380;enie algebraiczne, w kt&#243;rym ka&#380;dy wyraz jest sta&#322;&#261; lub iloczynem sta&#322;ej i jednej zmiennej. R&#243;wnania te przedstawiaj&#261; linie proste po naniesieniu na uk&#322;ad wsp&#243;&#322;rz&#281;dnych Kartezjusza, odzwierciedlaj&#261;c sta&#322;e tempo zmiany mi&#281;dzy dwiema wielko&#347;ciami.Typowe r&#243;wnanie liniowe z jedn&#261; zmienn&#261; ma posta&#263;: ax + b = c, gdzie a, b i c to sta&#322;e, a x to zmienna. Proces rozwi&#261;zywania polega na wyodr&#281;bnieniu zmiennej w celu okre&#347;lenia jej warto&#347;ci, dzi&#281;ki czemu r&#243;wnanie jest spe&#322;nione. Odbywa si&#281; to poprzez operacje takie jak dodawanie, odejmowanie, mno&#380;enie lub dzielenie, z zachowaniem r&#243;wno&#347;ci po obu stronach r&#243;wnania.R&#243;wnania liniowe s&#261; cz&#281;sto wykorzystywane do modelowania scenariuszy finansowych, takich jak okre&#347;lanie struktury koszt&#243;w. Rozwa&#380;my firm&#281; taks&#243;wkarsk&#261;, kt&#243;ra pobiera sta&#322;&#261; op&#322;at&#281; oraz sta&#322;&#261; stawk&#281; za kilometr. Je&#347;li 10-kilometrowy przejazd kosztuje 35 USD, a sta&#322;a op&#322;ata jest znana, nieznan&#261; stawk&#281; za kilometr mo&#380;na przedstawi&#263; jako r&#243;wnanie liniowe. Oznaczmy stawk&#281; za kilometr przez r, a sta&#322;a op&#322;ata wynosi 5 USD. R&#243;wnanie wygl&#261;da nast&#281;puj&#261;co:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40030/40030_Equation_1_1__editor_40030.svg' alt='40030_Equation_1_1_.svg' /></figure>Odejmuj&#261;c sta&#322;&#261; op&#322;at&#281; i rozwi&#261;zuj&#261;c r&#243;wnanie wzgl&#281;dem r:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/450;' src='https://cloudfront.jove.com/files/ftp_upload/40030/40030_Equation_2_1__editor_40030.svg' alt='40030_Equation_2_1_.svg' /></figure>Zatem stawka za kilometr wynosi 3 USD. Ponowne podstawienie potwierdza rozwi&#261;zanie: 10 &#215; 3 + 5 = 35. Ten przyk&#322;ad ilustruje, jak r&#243;wnania liniowe s&#261; niezb&#281;dnymi narz&#281;dziami do kwantyfikacji zale&#380;no&#347;ci i efektywnego rozwi&#261;zywania problem&#243;w praktycznych.

Równania liniowe

R&#243;wnania liniowe stanowi&#261; podstaw&#281; wielu zastosowa&#324; algebraicznych i rzeczywistych, charakteryzuj&#261;c si&#281; prostot&#261; i ...

Education

JoVE Core

Math Fundamentals

Podstawy matematyki

Unit

Jednostka

Foundations of Mathematics

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

Linear Equations

solving linear equations

Linear equations are a basic algebra tool for modeling constant rates and everyday relationships. They use constants and a single variable, and each term is either a constant or a product of a constant and that variable. When graphed on a Cartesian coordinate plane, they form a straight line that shows a constant rate of change between two quantities.

A linear equation in one variable often has the form ax + b = c, where a, b, and c are constants and x is the variable. Solving the equation means isolating the variable so the statement becomes true. To do that, you use addition, subtraction, multiplication, or division while keeping both sides equal.

These equations are useful in real-world settings, especially for money problems. A cab company can charge a flat fee plus a fixed cost per kilometer. If a 10-kilometer ride costs $35 and the flat fee is $5, the per-kilometer rate can be found with a linear equation.

Let the rate per kilometer be r. Subtract the flat fee from the total, then solve for r. The result is $3 per kilometer. Checking the solution gives 10 × 3 + 5 = 35, which confirms the equation works. This kind of example shows how linear equations help students describe relationships and solve practical problems efficiently.

Solving Linear Equations

real number system

Number Types on the Real Line

The real number system brings together the number types used for counting, measuring, and comparing quantities. It includes natural numbers, integers, rational numbers, and irrational numbers. Each set adds new kinds of numbers to the same system. Natural numbers are the basic counting numbers: 1, 2, 3, and so on. Integers extend that set by adding zero and negative whole numbers, such as ..., -3, -2, -1, 0, 1, 2, .... Rational numbers are numbers that can be written as a ratio of two...

real number operations

Real Numbers on the Number Line

Real numbers are the values that can be placed on a continuous number line. The system grew from counting numbers used for enumeration. It later expanded to include zero, which shows the absence of quantity, and negative numbers, which are opposites of counting numbers. The real number system also includes fractions and decimals that represent parts of a whole or equal divisions. When written in decimal form, these numbers may end or repeat in a regular pattern. Together, these values describe...

exponents

Exponent Rules and Scientific Notation

Exponents show repeated multiplication in a short form. They are important in algebra and in many other math topics, including scientific computation, scaling laws, and dimensional analysis. For a nonzero real number a and an integer n, a^n means a multiplied by itself n times. Exponent rules help simplify expressions with powers. These properties make it easier to work with exponential expressions in both symbolic and numerical form. They also support common calculations that use exponents in...

radicals

Simplifying Radicals

Radicals show the root of a number and the exponent needed to get that value. A radical expression has two main parts. The radicand is the number inside the root symbol, and the index tells which root to take. The notation n√a means the principal nth root of a. The index changes the type of root. When n equals 2, the expression is a square root. When n equals 3, it is a cube root. The sign of the radicand also matters. If the index is even, a negative radicand does not give a real result. If...

algebraic expressions

Simplifying Algebraic Expressions

Algebraic expressions use variables, constants, and operations to show mathematical relationships. They help describe patterns and solve problems across many areas of mathematics. Knowing the parts of an expression makes it easier to simplify and work with it correctly. Each algebraic expression has separate pieces that work together. A coefficient is the number attached to a variable, and an exponent shows repeated multiplication. Some expressions have a single part, while others include...

rational expressions

Working with Rational Expression Domains

Rational expressions are algebraic fractions with polynomials in the numerator and denominator. They follow the same basic rules as numerical fractions, but variables make them more delicate to work with. A key step is finding values that make the expression undefined, usually because of division by zero or an undefined radical. To determine the domain, start by identifying the values excluded from the denominator. The domain includes all real numbers except those that make the denominator...

linear equations

Linear equations are a basic algebra tool for modeling constant rates and everyday relationships. They use constants and a single variable, and each term is either a constant or a product of a constant and that variable. When graphed on a Cartesian coordinate plane, they form a straight line that shows a constant rate of change between two quantities. A linear equation in one variable often has the form ax + b = c, where a, b, and c are constants and x is the variable. Solving the equation...

quadratic equations

Solving Quadratics by Factoring and Formula

Quadratic equations are algebraic equations with a variable squared, a first-power term, and a constant term. They are set equal to zero. In high school math, they often help model area, motion, and optimization problems. The general form of a quadratic equation uses real numbers a, b, and c, with a not equal to zero. The nonzero a value keeps the squared term in the equation. One common way to solve a quadratic is to rewrite it as a product of two linear expressions. In factored form, the...

complex numbers

Complex Numbers in Solving Equations

Complex numbers extend the real number system so equations with negative square roots can still be solved. The real numbers cannot represent the square root of a negative number. That limits some quadratic equations, especially those with negative discriminants. To solve this problem, mathematicians introduced the imaginary unit i, where i = √(-1). A complex number is written as x + yi, where x and y are real numbers. In this form, x is the real part and y is the imaginary part. The key rule...

quadratic equations in the complex number system

quadratic-equations-in-the-complex-number-system

Complex Roots of Quadratic Equations

Quadratic equations can have real or complex roots, depending on the discriminant. For an equation in the form ax² + bx + c = 0, the coefficients a, b, and c are the numbers that multiply x², x, and the constant term. The discriminant, b² − 4ac, tells us what kind of solutions to expect. When the discriminant is negative, the equation has no real number solutions. In that case, complex numbers provide a way to continue solving the problem. The imaginary unit i is defined by i = √−1, so a...

radical equations

Solving Radical Equations with Checks

Radical equations are equations with a variable inside a radical, most often a square root or cube root. These equations appear in science, engineering, and real-world measurement problems that involve nonlinear relationships. Solving them starts by isolating the radical expression. Then the radical is removed by raising both sides to the power that matches the radical’s index. This process can create extraneous solutions, which are values that work in the transformed equation but not in the...

inequalities

Solving Inequalities and Interval Notation

Inequalities show how two values compare when they are not equal. They use symbols such as , ≤, and ≥. These statements often describe a range of possible answers instead of one exact value. Interval notation is a shorter way to write these solution sets. An open interval, written as (a, b), leaves out the endpoints. A closed interval, written as [a, b], includes both endpoints. Half-open intervals, such as (a, b] and [a, b), include only one endpoint. Intervals can also extend forever in one...

absolute value inequalities

Absolute Value Inequalities on a Number Line

Absolute value inequalities show how far a number can be from zero or from a target value. Absolute value means distance on the number line, so the sign of the number does not matter. In inequalities, this distance helps define a valid range of values for a variable. An inequality of the form |x| ≤ a, where a ≥ 0, includes every x between −a and a. This can also be written as a compound inequality. On a number line, the solution is shown as a solid segment from −a to a with closed circles at...

mathematical modeling: problem solving

Mathematical Modeling for Loan Payments

Mathematical modeling turns a real-life situation into an equation that can be solved. It starts with defining the problem, choosing measurable quantities, and assigning variables to them. Then a matching model is selected and the equation is solved for the unknown value. This process is especially useful in finance, where models help evaluate investments, loans, and repayment plans. One common example is finding the fixed monthly payment on a loan. That calculation uses the standard annuity...

key terms and concepts

Learn how linear equations use a variable, constants, and simple operations to solve real-world problems like cab fare rates and constant change.

Solving Linear Equations | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Solving Linear Equations

Description

Transcript

Tags