Solve systems of linear equations by using the linear combinations method, Solve pairs of linear equations using patterns, Solve linear systems algebraically using substitution. x Find the numbers. = 3 Some students who correctly write \(2m-2(2m+10)=\text-6\) may fail to distribute the subtraction and write the left side as\(2m-4m+20\). Which method do you prefer? 1 Determine whether the ordered pair is a solution to the system: \(\begin{cases}{xy=1} \\ {2xy=5}\end{cases}\). If you missed this problem, review Example 1.123. {2x+y=11x+3y=9{2x+y=11x+3y=9, Solve the system by substitution. x 2 Since 0 = 10 is a false statement the equations are inconsistent. y x Solve a system of equations by substitution, Solve applications of systems of equations by substitution. { We begin by solving the first equation for one variable in terms of the other. \\ + y x 8 x & - & 6 y & = & -12 This means Sondra needs 2 quarts of club soda and 8 quarts of fruit juice. 15, { Since both equations are solved for y, we can substitute one into the other. + Step 2. Look at the system we solved in Exercise \(\PageIndex{19}\). + + = \end{array}\nonumber\], To find \(x,\) we can substitute \(y=1\) into either equation of the original system to solve for \(x:\), \[x+1=7 \quad \Longrightarrow \quad x=6\nonumber\]. The sum of two numbers is 30. x The perimeter of a rectangle is 58. Uh oh, it looks like we ran into an error. by substitution. = -5 x &=-30 \quad \text{subtract 70 from both sides} \\ Coincident lines have the same slope and same y-intercept. 3 Solve the system by substitution. + Well organize these results in Figure \(\PageIndex{2}\) below: Parallel lines have the same slope but different y-intercepts. \end{array}\right)\nonumber\]. << /Length 16 0 R /Filter /FlateDecode /Type /XObject /Subtype /Form /FormType -3 x & + & 2 y & = & 3 \\ \end{array}\nonumber\]. 5 x & + & 10 y & = & 40 It has no solution. {x+y=44xy=2{x+y=44xy=2. The graph of a linear equation is a line. Step 5. = Step 3. = y x Find the measure of both angles. Rearranging or solving \(4+ y=12\) to get \(y =8\), and then substituting 8 for \(y\) in the equation\(y=2x - 7\): \(\begin {align} y&=2x - 7\\8&=2x - 7\\ 15&=2x \\ 7.5 &=x\end{align}\). << /Length 8 0 R /Filter /FlateDecode /Type /XObject /Subtype /Form /FormType y = How many cars would need to be sold to make the total pay the same? In this section we solve systems of two linear equations in two variables using the substitution method. 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. Make the coefficients of one variable opposites. x then you must include on every digital page view the following attribution: Use the information below to generate a citation. We will graph the equations and find the solution. = + Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions. No labels or scale. 3 2 The length is 5 more than the width. To illustrate this, let's look at Example 27.3. Unit test Test your knowledge of all skills in this unit. y Determine whether the lines intersect, are parallel, or are the same line. 3 1 Before we are truly finished, we should check our solution. + = x = Step 3. In the section on Solving Linear Equations and Inequalities we learned how to solve linear equations with one variable. 1 Graph the second equation on the same rectangular coordinate system. To solve for x, first distribute 2: Step 4: Back substitute to find the value of the other coordinate. x Make sure students see that the last two equations can be solved by substituting in different ways. 16, { 5 3 1 = Solve each system. + 3 The length is 10 more than three times the width. Maxim has been offered positions by two car dealers. y 3.8 -Solve Systems of Equations Algebraically (8th Grade Math)All written notes and voices are that of Mr. Matt Richards. One number is 10 less than the other. All four systems include an equation for either a horizontal or a vertical line. y For example, 3x + 2y = 5 and 3x. 2 We also categorize the equations in a system of equations by calling the equations independent or dependent. Solve a System of Equations by Substitution We will use the same system we used first for graphing. endstream x Solution To Lesson 16 Solve System Of Equations Algebraically Part I You Solving Systems Of Equations Algebraically Examples Beacon Lesson 16 Solve Systems Of Equations Algebraically Ready Common Core Solving Systems Of Equations Algebraiclly Section 3 2 Algebra You Warrayat Instructional Unit y 3 = The activity allows students to practicesolving systems of linear equations by substitution and reinforces the idea thatthere are multiple ways to perform substitution. y 7, { Khan Academy is a 501(c)(3) nonprofit organization. + y + { 2 y endobj Substitute the expression from Step 1 into the other equation. y \[\begin{cases}{2 x+y=7} \\ {x-2 y=6}\end{cases}\]. (4, 3) is a solution. Doing thisgives us an equation with only one variable, \(p\), and makes it possible to find\(p\). 4 = 7 y 11. = 16 Does a rectangle with length 31 and width. \(\begin{cases}{y=2x4} \\ {4x+2y=9}\end{cases}\), \(\begin{cases}{y=\frac{1}{3}x5} \\ {x-3y=6}\end{cases}\), Without graphing, determine the number of solutions and then classify the system of equations: \(\begin{cases}{2x+y=3} \\ {x5y=5}\end{cases}\), \(\begin{array}{lrrlrl} \text{We will compare the slopes and intercepts} & \begin{cases}{2x+y=-3} \\ {x5y=5}\end{cases} \\ \text{of the two lines.} & 6 x+2 y=72 \\ Here are graphs of two equations in a system. 2. use algebraic techniques to solve a system of linear equations in two variables, in particular the elimination method and substitution; 3. determine efficient or elegant approaches to finding a solution to a system of linear equations in two variables 4. relate an algebraic solution to a system of equations in two variables to a graphical Without technology, however, it is not easy to tell what the exact values are. y y 8 = 1 = + y y Step 3: Solve for the remaining variable. The point of intersection (2, 8) is the solution. x x 2 x endobj = 1 /BBox [18 40 594 774] /Resources 13 0 R /Group << /S /Transparency /CS 14 0 R Find the measure of both angles. If the equations are given in standard form, well need to start by solving for one of the variables. = \Longrightarrow & 2 y=-6 x+72 & \text{subtract 6x from both sides} \\ 2 Think about this in the next examplehow would you have done it with just one variable? = y = 4 The perimeter of a rectangle is 84. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo Lesson 2: 16.2 Solving x^2 + bx + c = 0 by Factoring . = Substitute the expression found in step 1 into the other equation. = We say the two lines are coincident. {x6y=62x4y=4{x6y=62x4y=4. 4 xYGrSX>EX0]x!j8h^VDfeVn~3###%5%M)7e y (In each of the first three systems, one equation is already in this form. = Finally, we check our solution and make sure it makes both equations true. 4 8 A system of two linear equations in two variables may have one solution, no solutions, or infinitely many solutions. Hence, we get \(x=6 .\) To find \(y,\) we substitute \(x=6\) into the first equation of the system and solve for \(y\) (Note: We may substitute \(x=6\) into either of the two original equations or the equation \(y=7-x\) ): \[\begin{array}{l} 8 x = Find the measure of both angles. y 4 Since the least common multiple of 2 and 3 is \(6,\) we can multiply the first equation by 3 and the second equation by \(2,\) so that the coefficients of \(y\) are additive inverses: \[\left(\begin{array}{lllll} We recommend using a + \end{align*}\right)\nonumber\]. Exercise 3. Record and display their responses for all to see. y 2 15 0 obj Solve the system. y Consider asking students to usesentence starters such as these: With a little bit of rearrangement, allsystems could be solved by substitution without cumbersome computation, but system 2 would be most conducive to solving by substitution. 3 y A student has some $1 bills and $5 bills in his wallet. If some students struggle with the last system because the variable that is already isolated is equal to an expression rather than a number, askwhat they would do if the first equation were \(y= \text{a number}\)instead of \(y=2x-7\). = Theequations presented andthereasoning elicited here will be helpful later in the lesson, when students solve systems of equations by substitution. 3 5, { {y=x+10y=14x{y=x+10y=14x. The latter has a value of 13,not 20.). 2 Lesson 13 Solving Systems of Equations; Lesson 14 Solving More Systems; Lesson 15 Writing Systems of Equations; Let's Put It to Work. = x 3 aF s|[ RS9&X110!fH:dfeTisGR% 33-u6D,+i6fu2tzm%Ll[0,p uBEs7bS15a;m8n``s xqLZ335,C`m ~9["AnySNR~6jedyhg/`gIn&Y2y y=J(?%$oXBsjb7:=o3c1]bsv^jFahLScN{qQHv(vj"z,4A$8sCDcc4Hn*F+Oi8?DurqJ32!?D_oc)q/NE~'q+s9M#~Aas;Q(" P>CIwj^fnGdzm0%.+pjsGf:M?9iT^KHnTpd5y When both equations are already solved for the same variable, it is easy to substitute! \(\begin{cases}{ f+c=10} \\ {f=4c}\end{cases}\). = = 15 The length is 5 more than three times the width. x 4 x & - & 3 y & = & -6 (4, 3) does not make both equations true. y 14 To illustrate, we will solve the system above with this method. {2xy=1y=3x6{2xy=1y=3x6. Monitor for the different ways that students use substitutions to solve the systems. The equations have coincident lines, and so the system had infinitely many solutions. = y 4 8 Step 4. The second equation is already solved for \(y\) in terms of \(x\) so we can substitute it directly into \(x+y=1\) : \[x+(-x+2)=1 \Longrightarrow 2=1 \quad \text { False! These are called the solutions to a system of equations. 3 y 6, { 3 11, Solve Applications of Systems of Equations by Substitution. = y Legal. 2 For instance, given a system with \(x=\text-5\) as one of the equations, they may reason that any point that has a negative \(x\)-valuewill be to the left of the vertical axis. x+y=7 \Longrightarrow 6+1=7 \Longrightarrow 7=7 \text { true! } = 10 Find the intercepts of the second equation. y \(\begin{cases}{3x2y=4} \\ {y=\frac{3}{2}x2}\end{cases}\), \(\begin{array}{lrrlrl} \text{We will compare the slopes and intercepts of the two lines. y = For example: To emphasize that the method we choose for solving a systems may depend on the system, and that somesystems are more conducive to be solved by substitution than others, presentthe followingsystems to students: \(\begin {cases} 3m + n = 71\\2m-n =30 \end {cases}\), \(\begin {cases} 4x + y = 1\\y = \text-2x+9 \end {cases}\), \(\displaystyle \begin{cases} 5x+4y=15 \\ 5x+11y=22 \end{cases}\). 1 Half an hour later, Tina left Riverside in her car on the same route as Stephanie, driving 70 miles per hour. = + & 3 x+8 y=78 \\ This Math Talk encourages students to look for connections between the features of graphsandof linear equations that each represent a system. 2 = x Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. x Be prepared to explain how you know. 4 5 x Solution: First, rewrite the second equation in standard form. Solve the resulting equation. HOW TO SOLVE A SYSTEM OF EQUATIONS BY ELIMINATION. y Substitute the solution in Step 3 into one of the original equations to find the other variable. Solve the system of equations{x+y=10xy=6{x+y=10xy=6. 4 0 obj 2 \\ \text{Write the second equation in} \\ \text{slopeintercept form.} Solve a system of equations by substitution. = y = y x 15 x {5x+2y=124y10x=24{5x+2y=124y10x=24. s"H7:m$avyQXM#"}pC7"q$:H8Cf|^%X 6[[$+;BB^ W|M=UkFz\c9kS(8<>#PH` 9 G9%~5Y, I%H.y-DLC$a, $GYE$ y Select previously identified students to share their responses and reasoning. When two or more linear equations are grouped together, they form a system of linear equations. The measure of one of the small angles of a right triangle is 26 more than 3 times the measure of the other small angle. 06x! = y y x Solve for yy: 8y8=322y8y8=322y \end{array}\right)\nonumber\], \[-1 x=-3 \quad \Longrightarrow \quad x=3\nonumber\], To find \(y,\) we can substitute \(x=3\) into the first equation (or the second equation) of the original system to solve for \(y:\), \[-3(3)+2 y=3 \Longrightarrow-9+2 y=3 \Longrightarrow 2 y=12 \Longrightarrow y=6\nonumber\]. = We will first solve one of the equations for either x or y. {x+3y=104x+y=18{x+3y=104x+y=18. x Solving a System of Two Linear Equations in Two Variables using Elimination Multiply one or both equations by a nonzero number so that the coefficients of one of the variables are additive inverses. x Manny needs 3 quarts juice concentrate and 9 quarts water. 2 2 y y x 4, { Want to cite, share, or modify this book? The ordered pair (3, 2) made one equation true, but it made the other equation false. 2 y = The number of quarts of fruit juice is 4 times the number of quarts of club soda. + 1 8 0 obj Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. {3x+y=52x+4y=10{3x+y=52x+4y=10. Geraldine has been offered positions by two insurance companies. Solve the system of equations{3x+y=12x=y8{3x+y=12x=y8 by substitution and explain all your steps in words. x+TT(T0 B3C#sK#Tp}\C|@ y \(\begin {cases} 3p + q = 71\\2p - q = 30 \end {cases}\). Solve a system of equations by substitution. ^1>}{}xTf~{wrM4n[;n;DQ]8YsSco:,,?W9:wO\:^aw 70Fb1_nmi!~]B{%B? ){Cy1gnKN88 7=_`xkyXl!I}y3?IF5b2~f/@[B[)UJN|}GdYLO:.m3f"ZC_uh{9$}0M)}a1N8A_1cJ j6NAIp}\uj=n`?tf+b!lHv+O%DP$,2|I&@I&$ Ik I(&$M0t Ar wFBaiQ>4en; Some students may rememberthat the equation for such lines can be written as \(x = a\) or\(y=b\), where \(a\) and \(b\)are constants. = An example of a system of two linear equations is shown below. 12 By the end of this section, you will be able to: Before you get started, take this readiness quiz. {5x3y=2y=53x4{5x3y=2y=53x4. See the image attribution section for more information. x Display their work for all to see. Its graph is a line. = + \(\begin{cases}{4x5y=20} \\ {y=\frac{4}{5}x4}\end{cases}\), infinitely many solutions, consistent, dependent, \(\begin{cases}{ 2x4y=8} \\ {y=\frac{1}{2}x2}\end{cases}\). 2 Without graphing, determine the number of solutions and then classify the system of equations. y If the graphs extend beyond the small grid with x and y both between 10 and 10, graphing the lines may be cumbersome. x Choose variables to represent those quantities. Line 1 starts on vertical axis and trends downward and right. For full sampling or purchase, contact an IMCertifiedPartner: \(\begin{cases} 3x = 8\\3x + y = 15 \end{cases} \), \(\begin{cases}3 x + 2y - z + 5w= 20 \\ y = 2z-3w\\ z=w+1 \\ 2w=8 \end{cases}\), \(\begin {align} 3(20.2) + q &=71\\60.6 + q &= 71\\ q &= 71 - 60.6\\ q &=10.4 \end{align}\), Did anyone have the same strategy but would explain it differently?, Did anyone solve the problem in a different way?. Translate into a system of equations. }{=}}&{0} \\ {-1}&{=}&{-1 \checkmark}&{0}&{=}&{0 \checkmark} \end{array}\), \(\begin{aligned} x+y &=2 \quad x+y=2 \\ 0+y &=2 \quad x+0=2 \\ y &=2 \quad x=2 \end{aligned}\), \begin{array}{rlr}{x-y} & {=4} &{x-y} &{= 4} \\ {0-y} & {=4} & {x-0} & {=4} \\{-y} & {=4} & {x}&{=4}\\ {y} & {=-4}\end{array}, We know the first equation represents a horizontal, The second equation is most conveniently graphed, \(\begin{array}{rllrll}{y}&{=}&{6} & {2x+3y}&{=}&{12}\\{6}&{\stackrel{? \\ &3x-2y&=&4 \\ & -2y &=& -3x +4 \\ &\frac{-2y}{-2} &=& \frac{-3x + 4}{-2}\\ &y&=&\frac{3}{2}x-2\\\\ \text{Find the slope and intercept of each line.}

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