Name: plotting two-variable equations with tables
Uploaded: 2025-11-26T05:44:21.000Z
Duration: 1 min 30 s
Channel: JoVE

Une équation à deux variables montre une relation entre x et y. Les solutions de l’équation sont des paires ordonnées de valeurs, (x, y), qui satisfont l’équation.Ces solutions peuvent être représentées visuellement en traçant les points correspondants sur un graphique.Le processus de représentation graphique commence par la sélection de quelques valeurs pour x et leur substitution dans l’équation pour calculer les valeurs correspondantes de y.Les points à l’origine identifient l’endroit où le graphique croise les axes. L’ordonnée à l’origine est trouvée en mettant y à zéro, et l’ordonnée à l’origine est trouvée en mettant x à zéro.Ces points sont ensuite ajoutés au graphique avec les autres points tracés.Chaque paire ordonnée résultante est enregistrée et tracée sur un graphique. Les points sont ensuite reliés pour révéler un motif visuel, qui peut apparaître sous la forme d’une ligne droite ou d’une courbe, selon l’équation.Cette méthode graphique est utile dans des contextes réels, comme le suivi des ventes de billets de cinéma sur plusieurs jours.Le graphique montre comment les ventes évoluent au fil du temps en traçant les ventes de billets quotidiens sous forme de points où x est le numéro du jour et y est le total des ventes. Ce modèle visuel permet de comparer les valeurs, de repérer les tendances et de prédire les ventes futures en fonction des données passées.

représentations graphiques des équations à deux variables

Une &#233;quation &#224; deux variables, g&#233;n&#233;ralement exprim&#233;e sous la forme y = f(x) ou Ax + By = C, d&#233;crit la relation entre deux grandeurs repr&#233;sent&#233;es par x et y. Chaque solution d&#8217;une telle &#233;quation correspond &#224; un couple ordonn&#233; (x, y) qui satisfait l&#8217;&#233;quation une fois les valeurs substitu&#233;es. Ces couples peuvent &#234;tre repr&#233;sent&#233;s graphiquement afin de visualiser la relation entre les variables. 
Une m&#233;thode courante pour tracer la repr&#233;sentation graphique d&#8217;une &#233;quation &#224; deux variables consiste &#224; &#233;tablir un tableau de valeurs. On commence par choisir plusieurs valeurs de la variable ind&#233;pendante x, puis on substitue chacune d&#8217;elles dans l&#8217;&#233;quation pour calculer la valeur correspondante de y. Les couples (x, y) obtenus repr&#233;sentent des points du plan cart&#233;sien. En tra&#231;ant ces points puis en les reliant, on met en &#233;vidence la nature de la relation, qu&#8217;elle soit lin&#233;aire, quadratique ou d&#8217;un autre type.
Par exemple, l&#8217;&#233;quation lin&#233;aire y = 2x + 1 engendre une droite lorsque plusieurs valeurs de x sont utilis&#233;es pour d&#233;terminer les valeurs correspondantes de y&#160;:
<ul>
<li>Si x = -1, alors y = 2(-1) + 1 = -1</li>
<li>Si x = 0, alors y = 1</li>
<li>Si x = 2, alors y = 5</li>
</ul>
Une fois ces points (-1, -1), (0, 1) et (2, 5) trac&#233;s et reli&#233;s, ils forment une droite repr&#233;sentant l&#8217;ensemble des solutions de l&#8217;&#233;quation.
Les points d&#8217;intersection constituent des rep&#232;res essentiels pour comprendre le comportement du graphe. L&#8217;intersection avec l&#8217;axe des abscisses se d&#233;termine en posant y = 0 et en calculant x, tandis que l&#8217;intersection avec l&#8217;axe des ordonn&#233;es s&#8217;obtient en posant x = 0 et en calculant y. Ces points indiquent l&#8217;endroit o&#249; la courbe coupe les axes x et y. Pour l&#8217;&#233;quation y = 2x + 1, en posant x = 0, on obtient l&#8217;ordonn&#233;e &#224; l&#8217;origine (0, 1), et en posant y = 0, on obtient l&#8217;abscisse &#224; l&#8217;origine (&#8722;0,5, 0).

Représentations graphiques des équations à deux variables

Une &#233;quation &#224; deux variables, g&#233;n&#233;ralement exprim&#233;e sous la forme y = f(x) ou Ax + By = C, d&#233;crit la relation entre deux ...

Education

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JoVE Core

Math Fundamentals

Fondamentaux des mathématiques

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Unité

Coordinates and Graphs

Coordonnées et graphiques

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

Graphs of Equations in Two Variables

plotting two-variable equations with tables

Plotting two-variable equations with tables helps show how x and y are related. An equation written as y = f(x) or Ax + By = C gives ordered pairs that satisfy the rule. Each pair can be checked by substituting the x and y values into the equation.

A value table is a common way to build the graph. Start by choosing several values for the independent variable x. Then substitute each x value into the equation to find the matching y value. The results become ordered pairs on the Cartesian coordinate plane.

After the points are plotted, connect them to see the shape of the relationship. Some equations make a straight line, while others may form a quadratic or another curve. For example, the linear equation y = 2x + 1 gives points such as (-1, -1), (0, 1), and (2, 5).

Intercepts help identify where the graph crosses the axes. The x-intercept is found by setting y = 0 and solving for x. The y-intercept is found by setting x = 0 and solving for y. For y = 2x + 1, the y-intercept is (0, 1) and the x-intercept is (-0.5, 0).

Plotting Two-Variable Equations with Tables

plotting-two-variable-equations-with-tables

coordinate plane

Finding Points on the Coordinate Plane

The Cartesian coordinate plane is used to find and plot points in two dimensions. It is made of two number lines that cross at right angles. The horizontal line is the x-axis, and the vertical line is the y-axis. Their meeting point is the origin, where both values are zero. The axes divide the plane into four quadrants. These quadrants are labeled in a counterclockwise direction, starting from the upper right. Every point on the plane is written as an ordered pair. The first number is the...

the distance formula

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the midpoint formula

Finding the Center of a Line Segment

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graphs of equations in two variables

Plotting two-variable equations with tables helps show how x and y are related. An equation written as y = f(x) or Ax + By = C gives ordered pairs that satisfy the rule. Each pair can be checked by substituting the x and y values into the equation. A value table is a common way to build the graph. Start by choosing several values for the independent variable x. Then substitute each x value into the equation to find the matching y value. The results become ordered pairs on the Cartesian...

circles

Circle Equations in Standard Form

Circle equations in the coordinate plane use a center and a radius to describe every point on the curve. A circle is the set of all points that stay the same distance from a fixed point called the center. That constant distance is the radius. The distance formula connects a point x, y on the circle to the center h, k. The distance between those two points equals the radius r. When both sides of the distance formula are squared, the equation of the circle can be written in standard form. <br...

symmetry

Ellipse Symmetry at the Origin

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Graphing Equations to Find Intercepts

Graphing equations helps students find intercepts and solutions on the coordinate plane. It gives a visual way to solve equations and understand how a function behaves. To use this method, write the equation in the form y = f(x). The solution to the original equation is the x-value where the graph crosses the x-axis. At that point, f(x) = 0. For example, the linear equation 2x - 4 = 0 can be written as y = 2x - 4. When this function is graphed, it has one x-intercept at x = 2, and that is the...

solving inequalities graphically

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Graphing inequality solution ranges gives a visual way to find the x-values that make an inequality true. Students look at where a graph sits compared with the x-axis or with another graph. This shows the interval where the condition is satisfied. For a quadratic inequality, the first step is to plot the related function. For example, with x² − 4x + 2 &lt; 0, graph y = x² − 4x + 2. The solution is the set of x-values where the curve lies below the x-axis. Inequalities can also compare two...

key terms and concepts

Learn how to graph equations in two variables using value tables, ordered pairs, and intercepts for clear high school math practice.

Plotting Two-Variable Equations with Tables | 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

Plotting Two-Variable Equations with Tables

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