{"id":45,"date":"2017-08-23T07:10:40","date_gmt":"2017-08-23T07:10:40","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=45"},"modified":"2017-08-29T08:59:16","modified_gmt":"2017-08-29T08:59:16","slug":"solving-sytems-of-equations","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/solving-sytems-of-equations\/","title":{"rendered":"Solving Sytems of Equations"},"content":{"rendered":"
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\u2b05 Previous Lesson<\/a>\u00a0Workshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

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Solving Sytems of Equations<\/h1>\n

Objective<\/h4>\n

In this lesson, you will study how to solve systems of equations by graphing and substitution.<\/p>\n

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What are systems of equations?<\/h3>\n

An ordered pair<\/em> (x<\/em>,\u00a0y)<\/em> is the solution of a linear equation in two variables. A\u00a0linear equation<\/strong><\/em> may be graphed using two\u00a0points or one point and the slope.<\/p>\n

A set of two or more linear equations with the same variables\u00a0is a system<\/abbr> of linear equations. Sometimes these are called simultaneous\u00a0equations<\/strong>.<\/p>\n

If there are two equations and two variables, the set of all\u00a0ordered pairs that make both of the equations true is the solution\u00a0set<\/strong>. If there are three variables and three equations,\u00a0each solution is an ordered triple<\/strong>, such as (2, 4, 6).<\/p>\n

When you solve a system of equations, you are finding the\u00a0values for the variables that make all the equations true. If\u00a0there is at least one solution to the system, it is called a\u00a0consistent\u00a0system<\/abbr>. If there are an infinite number of solutions,\u00a0the system is consistent and the equations are called dependent\u00a0equations<\/abbr>. If the system has no solution, it is\u00a0called an inconsistent\u00a0system<\/abbr>.<\/p>\n

How do we use graphing to solve systems of equations?<\/h3>\n

To solve a system of equations by graphing, we graph all the\u00a0equations on a coordinate plane and find the point of\u00a0intersection.<\/p>\n

For example, solve the following system of equations by\u00a0graphing:<\/p>\n

<\/center>To graph these equations, we first solve for y<\/em>,\u00a0thereby putting the equations into slope-intercept\u00a0form<\/abbr>. The coefficient of x<\/em> will be the slope and the constant will be the y<\/em>-intercept.<\/p>\n

<\/center>The slope of the first line is 3, and the y<\/em>-intercept\u00a0is at the point (0, \u20139).<\/p>\n

<\/center>The slope of this line is\u00a0\u00a0and the y<\/em>-intercept\u00a0is at (0, 4).<\/p>\n

Graph these lines by plotting the y<\/em>-intercept\u00a0and then using the slope to find another point.<\/p>\n

\"Graph<\/center>The lines appear to intersect around the point (4, 3). To see\u00a0if we have solved the system, substitute these values back into\u00a0the original equations to see if they make the equations true.<\/p>\n

<\/center>Though this method accurately solved these equations, graphing\u00a0is not generally a very accurate way to solve systems of\u00a0equations.<\/p>\n

How do we use substitution to solve systems of equations?<\/h3>\n

A second, more accurate way to solve systems of equations is to\u00a0use substitution. To do this, we follow the following procedure.<\/p>\n

Step 1:<\/strong> Solve one of the equations for one of\u00a0the variables. This will give us the value of one variable in terms of another.<\/p>\n

Step 2:<\/strong> Substitute the expression we found for\u00a0that variable into the other equation.<\/p>\n

Step 3:<\/strong> Solve the equation into which the\u00a0expression has been substituted.<\/p>\n

Step 4:<\/strong> Using the value obtained, solve the\u00a0first equation.<\/p>\n

For example, solve the following system of equations by\u00a0substitution.<\/p>\n

<\/center>The first equation already has y<\/em> without a coefficient, so it will be easy to solve for y.<\/em><\/p>\n

<\/center>Then substitute 12 \u2013 2x<\/em> for y\u00a0<\/em>in the second equation and solve for x<\/em>.<\/p>\n

<\/center>Finally, substitute 5 for x\u00a0<\/em>in the first equation.<\/p>\n

<\/center>The ordered pair (5, 2) solves both equations, and you can\u00a0check by substituting into the original equations if you want to.<\/p>\n

Review of New Vocabulary and Concepts<\/h3>\n