lesson 16 solve systems of equations algebraically answer key

Lesson 1: 16.1 Solving Quadratic Equations Using Square Roots. y + The second equation is already solved for y. Find the length and width of the rectangle. = x 8 endobj y Some students may not remember to find the value of the second variable after finding the first. Highlight the strategies that involve substitution and name them as such. Solve the system by graphing: $\begin{cases}{y=\frac{1}{2}x3} \\ {x2y=4}\end{cases}$, Solve each system by graphing: $\begin{cases}{y=-\frac{1}{4}x+2} \\ {x+4y=-8}\end{cases}$, Solve each system by graphing: $\begin{cases}{y=3x1} \\ {6x2y=6}\end{cases}$, Solve the system by graphing: $\begin{cases}{y=2x3} \\ {6x+3y=9}\end{cases}$, Solve each system by graphing: $\begin{cases}{y=3x6} \\ {6x+2y=12}\end{cases}$, Solve each system by graphing: $\begin{cases}{y=\frac{1}{2}x4} \\ {2x4y=16}\end{cases}$. 2 We will focus our work here on systems of two linear equations in two unknowns. 4, { y 4 = Graph the second equation on the same rectangular coordinate system. Then try to . 6 x+2 y=72 \\ y & x+y=7 \\ \end{align*}\nonumber\]. 7, { 3 The first company pays a salary of $12,000 plus a commission of $100 for each policy sold. 2 $\begin {align} 2p - q &= 30 &\quad& \text {original equation} \\ 2p - (71 - 3p) &=30 &\quad& \text {substitute }71-3p \text{ for }q\\ 2p - 71 + 3p &=30 &\quad& \text {apply distributive property}\\ 5p - 71 &= 30 &\quad& \text {combine like terms}\\ 5p &= 101 &\quad& \text {add 71 to both sides}\\ p &= \dfrac{101}{5} &\quad& \text {divide both sides by 5} \\ p&=20.2 \end {align}$. 10 We will first solve one of the equations for either x or y. Rearranging or solving $4+ y=12$ to get $y =8$, and then substituting 8 for $y$ in the equation$y=2x - 7$: $\begin {align} y&=2x - 7\\8&=2x - 7\\ 15&=2x \\ 7.5 &=x\end{align}$. How many cable packages would need to be sold to make the total pay the same? = 2 = 4 = y 8 We will graph the equations and find the solution. y 9 x &=6 \quad \text{divide both sides by 5} This time, their job is to find a way to solve the systems. Step 2. Be prepared to explain how you know. Look back at the equations in Example 5.19. = y x The second pays a salary of $20,000 plus a commission of $25 for each cable package sold. 11 x 2 y y \end{array}\nonumber\]. 2 y OpenStax is part of Rice University, which is a 501(c)(3) nonprofit. Solve the system by graphing: $\begin{cases}{y=2x+1} \\ {y=4x1}\end{cases}$, Both of the equations in this system are in slope-intercept form, so we will use their slopes and y-intercepts to graph them. 1 7 Yes, 10 quarts of punch is 8 quarts of fruit juice plus 2 quarts of club soda. 1 5 + = y = 2 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. 1. 3 x y x endobj The length is 5 more than three times the width. x Answer the question with a complete sentence. = 4 Lesson 16 Vocabulary system of linear equations a set of two or more related linear equations that share the same variables . + Then we will substitute that expression into the other equation. + We will consider two different algebraic methods: the substitution method and the elimination method. 2 Step 1. { 2 Select previously identified students to share their responses and strategies. y \Longrightarrow & y=-3 x+36 & \text{divide both sides by 2} 1 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. $\begin{cases} x + 2y = 8 \\x = \text-5 \end{cases}$, $\begin{cases} y = \text-7x + 13 \\y = \text-1 \end{cases}$, $\begin{cases} 3x = 8\\3x + y = 15 \end{cases}$, $\begin{cases} y = 2x - 7\\4 + y = 12 \end{cases}$. The following steps summarize how to solve a system of equations by the elimination method: Solving a System of Two Linear Equations in Two Variables using Elimination, $\begin{array}{lllll} Quiz 1: 5 questions Practice what you've learned, and level up on the above skills. x & + &y & = & 7 \\ 12 The basic idea of the method is to get the coefficients of one of the variables in the two equations to be additive inverses, such as -3 and \(3,$ so that after the two equations are added, this variable is eliminated. Here is one way. y 1 When she spent 30 minutes on the elliptical trainer and 40 minutes circuit training she burned 690 calories. 2 We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. { coordinate algebra book lesson practice a 12 1 geometric sequences administration Mar 17 2022 web holt Step 4. The system has no solutions. The first company pays a salary of $ 14,000 plus a commission of $100 for each cable package sold. }{=}}&{4} \\ {2}&{=}&{2 \checkmark}&{4}&{=}&{4 \checkmark} \end{array}\), Solve each system by graphing: $\begin{cases}{x+y=6} \\ {xy=2}\end{cases}$, Solve each system by graphing: $\begin{cases}{x+y=2} \\ {xy=-8}\end{cases}$. Step 5. Does a rectangle with length 31 and width. Substitute the solution in Step 3 into one of the original equations to find the other variable. + = 11, Solve Applications of Systems of Equations by Substitution. x 3 3 In Example 5.19, it will take a little more work to solve one equation for x or y. When this is the case, it is best to first rearrange the equations before beginning the steps to solve by elimination. Simplify 42(n+5)42(n+5). = A system of equations that has at least one solution is called a consistent system. 8 Solve the system by substitution. There will be times when we will want to know how many solutions there will be to a system of linear equations, but we might not actually have to find the solution. But well use a different method in each section. Except where otherwise noted, textbooks on this site Find the x- and y-intercepts of the line 2x3y=12. \Longrightarrow & 2 y=-6 x+72 & \text{subtract 6x from both sides} \\ Translate into a system of equations. 6 + = x When both lines were in slope-intercept form we had: \[y=\frac{1}{2} x-3 \quad y=\frac{1}{2} x-2\]. + y The perimeter of a rectangle is 40. 4, { {4x3y=615y20x=30{4x3y=615y20x=30. 6 = The coefficients of the $x$ variable in our two equations are 1 and $5 .$ We can make the coefficients of $x$ to be additive inverses by multiplying the first equation by $-5$ and keeping the second equation untouched: \[\left(\begin{array}{lllll} 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: $\begin {cases} 3m + n = 71\\2m-n =30 \end {cases}$, $\begin {cases} 4x + y = 1\\y = \text-2x+9 \end {cases}$, $\displaystyle \begin{cases} 5x+4y=15 \\ 5x+11y=22 \end{cases}$. $\begin{cases} 5x 2y = 26 \\ y + 4 = x \end{cases}$, $\begin{cases} 2m 2p = \text-6\\ p = 2m + 10 \end{cases}$, $\begin{cases} 2d = 8f \\ 18 - 4f = 2d \end{cases}$, $\begin{cases} w + \frac17z = 4 \\ z = 3w 2 \end{cases}$, Solve this system with four equations.$\begin{cases}3 x + 2y - z + 5w= 20 \\ y = 2z-3w\\ z=w+1 \\ 2w=8 \end{cases}$, When solving the second system, students are likely tosubstitutethe expression $2m+10$ for $p$ in the first equation,$2m-2p=\text-6$. 6 5 The length is 10 more than three times the width. 5 0 obj endobj If two equations are dependent, all the solutions of one equation are also solutions of the other equation. How many ounces of coffee and how many ounces of milk does Alisha need? y=-x+2 x Well modify the strategy slightly here to make it appropriate for systems of equations. y 2 = + y Company B offers him a position with a salary of $24,000 plus a $50 commission for each stove he sells. Find step-by-step solutions and answers to Glencoe Math Accelerated - 9780076637980, as well as thousands of textbooks so you can move forward with confidence. Find the number of solutions to a system of equations (Eighth grade - Y.5), Classify a system of equations by graphing (Eighth grade - Y.6), Classify a system of equations (Eighth grade - Y.7), Solve a system of equations using substitution (Eighth grade - Y.8), Solve a system of equations using elimination (Eighth grade - Y.10).

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