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\(\left\{ \begin{aligned} y\amp=\frac{2}{3}x-4\\ y\amp=-2x+4\\ \end{aligned} \right.\)

The solution to the given system of equations is the ordered pair \((3,-2)\text{.}\)

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A linear equation in \(x\) and \(y\) is an equation that can be written in the form \(ax+by=c\) where \(a\text{,}\) \(b\text{,}\) and \(c\) are constants, and not both \(a\) and \(b\) are \(0\text{.}\) When graphed, a linear equation produces a line consisting of all of the ordered pairs that satisfy the equation (make the two sides of the equation have equal value).

A linear system of equations in two variables is a collection of two or more linear equations with those two variables. Solving a system of linear equations in two variables entails determining what ordered pairs, if any, satisfy every equation in the system. With the advent of graphing technology, one efficient way of determining the solution set to a system of linear equations is by graphing the line associated with every equation in the system and analyzing the result.

Consider the following system of equations. Please note that the large left brace is the symbol we use to indicate that the two equations form a system ā that we are looking for ordered pairs that satisfy both equations in the system.

\begin{equation*}
\left\{
\begin{aligned}
3x-2y\amp=-8\\
2x-y\amp=-6\\
\end{aligned}
\right.
\end{equation*}

If we were going to graph these two lines by hand, there would be value in manipulating the equations into slope-intercept from. Let'ahead and do that.

\begin{align*}
3x-2y\amp=-8\\
3x-2y\subtractright{3x}\amp=-8\subtractright{3x}\\
-2y\amp=-3x-8\\
\multiplyleft{-\frac{1}{2}}(-2y)\amp=\multiplyleft{-\frac{1}{2}}(-3x-8)\\
y\amp=\frac{3}{2}x+4
\end{align*}

\begin{align*}
2x-y\amp=-6\\
2x-y\subtractright{2x}\amp=-6\subtractright{2x}\\
-y\amp=-2x-6\\
\multiplyleft{(-y)}\amp=\multiplyleft{(-2x-6)}\\
y\amp=2x+6
\end{align*}

From the slope-intercept form of the line with equation \(3x-2x=-8\text{,}\) we can see that the line has a \(y\)-intercept of \((0,4)\) and a slope of \(\frac{3}{2}\text{.}\)

From the slope-intercept form of the line with equation \(2x-y=-6\text{,}\) we can see that the line has a \(y\)-intercept of \((0,6)\) and a slope of \(2\text{.}\)

As seen in FigureĀ 13.2.1, the two lines intersect at the point \((-4,-2)\text{.}\) So the solution to the given system of equations is the ordered pair \((-4,-2)\text{.}\)

When solving a system of two linear equations with two unknowns, there are three possible outcomes for the nature of the solution. The most common outcome is that there is exactly one ordered pair that satisfies both equations. This outcome is illustrated by the intersecting lines shown in FigureĀ 13.2.2 Another possible outcome is that no ordered pair satisfies both equations. This outcome is illustrated by the parallel lines shown in FigureĀ 13.2.3. The third possible outcome is that the two equations are equivalent ā that they are two manifestations of the equation for the same line. In this case, every point on the line satisfies both equations in the system. This outcome is illustrated by the single line shown in FigureĀ 13.2.4.

When there is at least one ordered pair that satisfies both equations in the system, the system is said to be consistent (FigureĀ 13.2.5 and FigureĀ 13.2.7). When there are no ordered pairs that satisfy both equations in the system, the system is said to be inconsistent (FigureĀ 13.2.6).

When there is exactly one or zero ordered pairs that satisfy both equations in the system, the equations in the system are said to be independent (FigureĀ 13.2.5 and FigureĀ 13.2.6). When there is more than one ordered pair that satisfies both equations in the system, the equations are said to be dependent (FigureĀ 13.2.7).

Solve each system of equations by graphing.

\(\left\{ \begin{aligned} y\amp=\frac{2}{3}x-4\\ y\amp=-2x+4\\ \end{aligned} \right.\)

Solution
Figure13.2.8\(\highlightr{y=\frac{2}{3}x-4}\) and \(\highlight{y=-2x+4}\)

The solution to the given system of equations is the ordered pair \((3,-2)\text{.}\)

\(\left\{ \begin{aligned} x+2y\amp=-5\\ -x+2y\amp=1\\ \end{aligned} \right.\)

Solution
Figure13.2.9\(\highlightr{x+2y=-5}\) and \(\highlight{-x+2y=1}\)

The solution to the given system of equations is the ordered pair \((-3,-1)\text{.}\)

\(\left\{ \begin{aligned} 5x-4y\amp=8\\ y\amp=\frac{5}{4}x+3\\ \end{aligned} \right.\)

Solution
Figure13.2.10\(\highlightr{5x-4y=8}\) and \(\highlight{y=\frac{5}{4}x+3}\)

The given system of equations has no solutions, it is inconsistent.