Name: solving linear equations by substitution
Uploaded: 2025-11-26T11:53:51.000Z
Duration: 1 min 25 s
Channel: JoVE

An airplane trip between two countries presents a classic puzzle for solving a system of equations.The distance is 1260 kilometers. Flying against the wind takes 3 hours, while flying with the wind takes only 2 hours.Although both trips cover the same distance, the time differs due to the wind&#8217;s speed, which remains constant throughout the journey.Let x be the airplane&#8217;s speed in still air, and y be the wind&#8217;s speed.Since distance equals speed multiplied by time, the effective speed against the wind is the airplane&#8217;s speed minus the wind&#8217;s speed. Multiplying this by 3 hours gives 1260 kilometers.For the journey with the wind, the effective speed is the airplane&#8217;s speed plus the wind&#8217;s speed. Multiplying this by 2 hours also gives 1260 kilometers.These two relationships form a pair of linear equations. Using the elimination method, adding the two equations removes the wind&#8217;s speed, leaving a simpler equation in x.This reveals the airplane&#8217;s true speed.Substituting this value into either equation gives the wind&#8217;s speed.

systems of linear equations in two variables

Systems of Linear Equations in Two Variables

Solving a system of linear equations is a fundamental concept in algebra. A system of equations consists of two or more linear equations involving the same set of variables. One of the most efficient algebraic methods for solving such systems is the substitution method. This technique involves expressing one variable in terms of the other from one equation and substituting it into the second equation. This method is particularly useful when one of the equations is easily rearranged.Consider the system of equations:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40068/40068_Equation_1_editor_40068.svg' alt='40068_Equation_1.svg' /></figure>First, we isolate one variable in one of the equations. From the second equation, solve for y:&#160;<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40068/40068_Equation_2_editor_40068.svg' alt='40068_Equation_2.svg' /></figure>Next, substitute the value of y into the first equation:<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/40068/40068_Equation_3_editor_40068.svg' alt='40068_Equation_3.svg' /></figure>Expand and simplify to find the value of x:<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/40068/40068_Equation_4_editor_40068.svg' alt='40068_Equation_4.svg' /></figure>Substitute x&#8203; into the expression for y:<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/40068/40068_Equation_4_editor_40068.svg' alt='40068_Equation_4.svg' /></figure>Since both equations are satisfied when substituting the calculated values of x and y, the solution is confirmed to be correct. Therefore, the solution to the system of equations is x = 27/14&#8203; and y = 19/7&#8203;. This means the pair represents the intersection point of the two lines described by the given equations. The substitution method has thus successfully provided a unique solution to the system.

Solving a system of linear equations is a fundamental concept in algebra. A system of equations consists of two or more linear equations involving the ...

systems-of-linear-equations-in-two-variables

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JoVE Core

Math Fundamentals

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Systems of Equations and Inequalities

KEY TERMS AND CONCEPTS

solving linear equations by substitution

Solving a system of linear equations by substitution is an important algebra skill. A system of equations uses two or more linear equations with the same variables. The substitution method is one efficient way to solve it. It works by writing one variable in terms of the other and then replacing that expression in the second equation.

This method is especially useful when one equation is easy to rearrange. In the example, one variable is isolated first. Then the second equation is solved for y. That expression for y is substituted into the first equation.

After substitution, the equation is expanded and simplified to find x. The value found for x is then put back into the expression for y. Both equations are checked to confirm the solution. The system has a unique solution, x = 27/14 and y = 19/7.

These values represent the point where the two lines intersect on a graph. The substitution method therefore gives the intersection point of the lines described by the equations. It provides a clear algebraic way to solve the system and confirm the result.

Solving Linear Equations by Substitution

systems of equations

Solving Systems by Graphing and Substitution

Systems of equations use multiple equations with the same variables. The goal is to find values that make every equation true at the same time. This approach helps compare several constraints or relationships within one problem. Three common ways to solve systems are substitution, elimination, and graphing. In substitution, you solve one equation for one variable and then replace that variable in the other equation. This reduces the system to a single-variable equation. In elimination, you...

Solving a system of linear equations by substitution is an important algebra skill. A system of equations uses two or more linear equations with the same variables. The substitution method is one efficient way to solve it. It works by writing one variable in terms of the other and then replacing that expression in the second equation. This method is especially useful when one equation is easy to rearrange. In the example, one variable is isolated first. Then the second equation is solved for y.

gaussian elimination: problem solving

Solving Linear Systems with Row Operations

Solving systems of linear equations often starts with row operations and Gaussian elimination. These systems use several variables, and each equation adds a linear constraint. A system with three equations and three variables can be written to show the relationship among x, y, and z. The goal is to find values that satisfy all three equations at the same time. Gaussian elimination simplifies the system into an equivalent upper triangular form. It uses elementary row operations to change the...

partial fractions

Decomposing Rational Expressions

Decomposing rational expressions into partial fractions helps turn a hard algebra problem into simpler pieces. A rational expression is a ratio of two polynomials. Partial fraction decomposition rewrites that ratio as a sum of easier fractions with simpler denominators, usually linear factors or irreducible quadratic factors. This method is useful when working with integration, solving equations, or analyzing algebraic expressions. It is a standard technique in algebra and calculus because it...

graphical representation of inequalities

Shading Inequality Graphs in the Coordinate Plane

Shading inequality graphs in the coordinate plane shows the region of points that satisfy a rule. A parabola, such as the graph of y = x squared, can act as the boundary. It divides the plane into different regions based on where points are located. When an equation changes to an inequality, the graph no longer shows just one curve. Signs such as greater than, less than, greater than or equal to, and less than or equal to create a shaded solution area instead. That shaded area represents all...

key terms and concepts

Learn how to solve a system of linear equations using substitution, find x and y, and confirm the intersection point of two lines.

Solving Linear Equations by Substitution | 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.

Solving Linear Equations by Substitution

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