Question

The following is a system of two linear equations in two variables. $\left\{\begin{array}{r}{x+y=1} \\ {2 x+2 y=2}\end{array}\right.$ The graph of the first equation is the same as the graph of the second equation, so the system has _ solutions. We express these solutions by writing $x=t$ $y=_$ where $t$ is any real number. Some of the solutions of this system are $(1,-),(-3,-),$ and $(5,-)$

Name: Problem 4, Chapter 10: Precalculus Mathematics for Calculus
Uploaded: 2020-05-12T19:47:36-08:00
Duration: 46 s
Channel: Jake Zanazzi
Description: The following is a system of two linear equations in two variables. { x+y=1 2 x+2 y=2 . The graph of the first equation is the same as the graph of the second equation, so the system has solutions. We express these solutions by writing x=t y= where t is any real number. Some of the solutions of this system are (1,-),(-3,-), and (5,-)

   The following is a system of two linear equations in two variables. $\left\{\begin{array}{r}{x+y=1} \\ {2 x+2 y=2}\end{array}\right.$ The graph of the first equation is the same as the graph of the second equation, so the system has _____ solutions. We express these solutions by writing $x=t$ $y=_____$ where $t$ is any real number. Some of the solutions of this system are $(1,-),(-3,-),$ and $(5,-)$

Precalculus Mathematics for Calculus

James Stewart,… 7th Edition

Chapter 10, Problem 4

Instant Answer

Solved by Expert Jake Zanazzi

05/12/2020

Step 1

The second equation is just the first equation multiplied by 2. This means that the graph of the first equation is the same as the graph of the second equation. Show more…

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The following is a system of two linear equations in two variables. $\left\{\begin{array}{r}{x+y=1} \\ {2 x+2 y=2}\end{array}\right.$ The graph of the first equation is the same as the graph of the second equation, so the system has _____ solutions. We express these solutions by writing $x=t$ $y=_____$ where $t$ is any real number. Some of the solutions of this system are $(1,-),(-3,-),$ and $(5,-)$

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Key Concepts

Dependent (Coincident) Equations

When two linear equations are multiples of each other, they describe the same line in the plane. This means the equations are dependent, and instead of having a single solution (where the lines intersect), they have an infinite number of solutions since every point on the line satisfies both equations.

Linear Equations in Two Variables

A linear equation in two variables represents a straight line on the coordinate plane. The solution to such an equation is the set of all points (x, y) that satisfy the equation, which geometrically corresponds to every point along the line.

Parameterization of Solutions

Parameterization involves expressing the solutions of an equation using a free parameter, often denoted by t. In the case of dependent linear equations, one variable is set equal to the parameter and the other is expressed in terms of the parameter, thereby capturing all the infinite solutions along the line.

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