Which ordered pair is a solution to the system of linear equations One-half x minus three-fourths y = StartFraction 11 Over 60 EndFraction and Two-fifths x + one-sixth y = StartFraction 3 Over 10 EndFraction?

<h2>Determining Whether An Ordered Pair is a Solution of a System of Equations</h2>
<p>In <a data-page="13" href="https://app.ulearngo.com/tutorials/solving-linear-equations-ii/">Solving Linear Equations and Inequalities</a> we learned how to solve linear equations with one variable. Remember that the solution of an equation is a value of the variable that makes a true statement when substituted into the equation.</p>
<p>Now we will work with <strong>systems of linear equations</strong>, two or more linear equations grouped together.</p>
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<h3>System of Linear Equations</h3>
<p>When two or more linear equations are grouped together, they form a system of linear equations.</p>
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<p>We will focus our work here on systems of two linear equations in two unknowns. Later, you may solve larger systems of equations.</p>
<p>An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations.</p>
<div>\(\begin{array}{c}2x+y=7\hfill \\ x-2y=6\hfill \end{array}\)</div>
<p>A linear equation in two variables, like 2<em>x</em> + <em>y</em> = 7, has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line.</p>
<p>To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pairs (<em>x</em>, <em>y</em>) that make both equations true. These are called the <strong>solutions to a system of equations</strong>.</p>
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<h3>Solutions of a System of Equations</h3>
<p><strong>Solutions of a system of equations</strong> are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair (<em>x</em>, <em>y</em>).</p>
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<p>To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.</p>
<p>Let’s consider the system below:</p>
<div>\(\begin{array}{c}3x-y=7\hfill \\ x-2y=4\hfill \end{array}\)</div>
<p>Is the ordered pair \(\left(2,-1\right)\) a solution?</p>
<p><img alt="This figure begins with a sentence, “We substitute x =2 and y = -1 into both equations.” The first equation shows that 3x minus y equals 7. Then 3 times 2 minus negative, in parentheses, equals 7. Then 7 equals 7 is true. The second equation reads x minus 2y equals 4. Then 2 minus 2 times negative one in parentheses equals 4. Then 4 = 4 is true." src="https://data.ulearngo.com/assets/37603c43-a3d0-4084-8412-b7b54e9332a6"/></p>
<p>The ordered pair (2, −1) made both equations true. Therefore (2, −1) is a solution to this system.</p>
<p>Let’s try another ordered pair. Is the ordered pair (3, 2) a solution?</p>
<p><img alt="This figure begins with the sentence, “We substitute x equals 3 and y equals 2 into both equations.” The first equation reads 3 times x minus 7equals 7. Then, 3 times 3 minus 2 equals 7. Then 7 = 7 is true. The second equation reads x minus 2y equals 4. The n times 2 minus 2 times 2 = 4. Then negative 2 = 4 is false." src="https://data.ulearngo.com/assets/90003f47-d1c4-421c-bfb5-544bd7c4627d"/></p>
<p>The ordered pair (3, 2) made one equation true, but it made the other equation false. Since it is not a solution to <strong>both</strong> equations, it is not a solution to this system.</p>
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<h2>Example</h2>
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<p>Determine whether the ordered pair is a solution to the system: \(\begin{array}{c}x-y=-1\hfill \\ 2x-y=-5\hfill \end{array}\)</p>
<p>a. \(\left(-2,-1\right)\)<br/>
b. \(\left(-4,-3\right)\)</p>
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<h3>Solution</h3>
<div>(a)</div>
<p><img alt="This figure shows two bracketed equations. The first is x minus y = negative 1. The second is 2 times x minus y equals negative 5. The sentence, “We substitute x = negative 2 and y = 1 into both equations,” follows. The first equation shows the substitution and reveals that negative 1 = negative 1. The second equation shows the substitution and reveals that 5 do not equal -5. Under the first equation is the sentence, “(negative 2, negative 1) does not make both equations true.” Under the second equation is the sentence, “(negative 2, negative 1) is not a solution.”" src="https://data.ulearngo.com/assets/d60ae624-2403-42ca-be3a-743449e7d050"/></p>
<div>(−2, −1) does not make both equations true. (−2, −1) is not a solution.</div>
<p>(b)</p>
<div><img alt="This figure begins with the sentence, “We substitute x = -4 and y = -3 into both equations.” The first equation listed shows x – y = -1. Then -4 - (-3) = -1. Then -1 = -1. The second equation listed shows 2x – y = -5. Then 2 times (-4) – (-3) = -5. Then -5 = -5. Under the first equation is the sentence, “(-4, -3) does make both equations true.” Under the second equation is the sentence, “(-4, -3) is a solution.”" src="https://data.ulearngo.com/assets/3057b4a3-b9c4-4b30-b45d-ea05c7161fd3"/></div>
<div>(−4, −3) does not make both equations true. (−4, −3) is a solution.</div>
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Determining Whether An Ordered Pair is a Solution of a System of Equations

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Determining Whether An Ordered Pair is a Solution of a System of Equations

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