Writing and solving equations in two variables assignment active

When an equation has two variables, any solution will be an ordered pair with a value for each variable.

Solution to a Linear Equation in Two Variables

An ordered pair [latex]\left(x,y\right)[/latex] is a solution of the linear equation [latex]ax+by=c[/latex], if the equation is a true statement when the [latex]x[/latex]– and [latex]y[/latex]-values of the ordered pair are substituted into the equation.

Example

Determine whether [latex](−2,4)[/latex] is a solution of the equation [latex]4y+5x=3[/latex].

Solution

Substitute [latex]x=−2[/latex] and [latex]y=4[/latex] into the equation: The statement is not true, so [latex](−2,4)[/latex] is not a solution.

Answer

[latex](−2,4)[/latex] is not a solution of the equation [latex]4y+5x=3[/latex].

example

Determine which ordered pairs are solutions of the equation [latex]x+4y=8\text<:>[/latex] 1. [latex]\left(0,2\right)[/latex] 2. [latex]\left(2,-4\right)[/latex] 3. [latex]\left(-4,3\right)[/latex]

Solution

Substitute the [latex]x\text<- and >y\text[/latex] from each ordered pair into the equation and determine if the result is a true statement.

1. [latex]\left(0,2\right)[/latex] 2. [latex]\left(2,-4\right)[/latex] 3. [latex]\left(-4,3\right)[/latex] [latex]x=\color, y=\color[/latex][latex]x+4y=8[/latex] [latex]\color+4\cdot\color\stackrel8[/latex] [latex]0+8\stackrel8[/latex] [latex]x=\color, y=\color[/latex][latex]x+4y=8[/latex] [latex]\color+4(\color)\stackrel8[/latex] [latex]2+(-16)\stackrel8[/latex] [latex]x=\color, y=\color[/latex][latex]x+4y=8[/latex] [latex]\color+4\cdot\color\stackrel8[/latex] [latex]-4+12\stackrel8[/latex] [latex]\left(0,2\right)[/latex] is a solution. [latex]\left(2,-4\right)[/latex] is not a solution. [latex]\left(-4,3\right)[/latex] is a solution.

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example

Determine which ordered pairs are solutions of the equation. [latex]y=5x - 1\text<:>[/latex] 1. [latex]\left(0,-1\right)[/latex] 2. [latex]\left(1,4\right)[/latex] 3. [latex]\left(-2,-7\right)[/latex]

Solution

Substitute the [latex]x\text<->[/latex] and [latex]y\text[/latex] from each ordered pair into the equation and determine if it results in a true statement.

1. [latex]\left(0,-1\right)[/latex] 2. [latex]\left(1,4\right)[/latex] 3. [latex]\left(-2,-7\right)[/latex] [latex]x=\color, y=\color[/latex][latex]y=5x-1[/latex] [latex]\color\stackrel5(\color)-1[/latex] [latex]-1\stackrel0-1[/latex] [latex]x=\color, y=\color[/latex][latex]y=5x-1[/latex] [latex]\color\stackrel5(\color)-1[/latex] [latex]4\stackrel5-1[/latex] [latex]x=\color, y=\color[/latex][latex]y=5x-1[/latex] [latex]\color\stackrel5(\color)-1[/latex] [latex]-7\stackrel-10-1[/latex] [latex]\left(0,-1\right)[/latex] is a solution. [latex]\left(1,4\right)[/latex] is a solution. [latex]\left(-2,-7\right)[/latex] is not a solution.

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The video shows more examples of how to determine whether an ordered pair is a solution of a linear equation.

Complete a Table of Solutions

In the previous examples, we substituted the [latex]x\text<- and >y\text[/latex] of a given ordered pair to determine whether or not it was a solution of a given linear equation. But how do we find the ordered pairs if they are not given? One way is to choose a value for [latex]x[/latex] and then solve the equation for [latex]y[/latex]. Or, choose a value for [latex]y[/latex] and then solve for [latex]x[/latex].

Let’s consider the equation [latex]y=5x - 1[/latex]. The easiest value to choose for [latex]x[/latex] or [latex]y[/latex] is zero:

We can continue to find more solutions by choosing different values of [latex]x[/latex] and [latex]y[/latex].