IM Alg1.2.14 Lesson: Solving Systems by Elimination (Part 1)
What do you notice? What do you wonder?
[table][tr][td][img]data:image/png;base64,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[/img][/td][td][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKkAAACICAYAAACcJcu8AAANmElEQVR4Ae2de1QU1x3H83fS054GNVaNCizs+wGICiguxooS1BglNSBW8IFRUUARBBUUUXzEECk+qCG+icEHaHy0KqYpGmtMTk48HtFq0n9iYzRpk7QRT1K/PRfP0l267O7M3JmdOL89Z8/u7Nx75/f73M+5szt3ZvYJ0IMIqJzAEyqPj8IjAiBJSQLVEyBJVd9FFCBJSg6ongBJqvouogBJUnJA9QRIUtV3EQVIkpIDqidAknLoonPnzqGuro5DS9SENwIkqTcqAj9jkj711FMCa1HxQAmQpIGS8lGOJPUBh8MqkpQDRJKUA0QfTZCkPuAEuookDZSUuHIkqThuHrVIUg8c3BdIUg5ISVIOEH00QZL6gBPoKpI0UFLiypGk4rh51CJJPXBwXyBJOSAlSTlA9NEESeoDTqCrSNJASYkrR5KK4+ZRiyT1wMF9gSTlgJQk5QDRRxMkqQ84ga4iSQMlJa4cSSqOm0ctktQDB/cFkpQDUpKUA0QfTZCkPuAEuookDZSUuHIkqThuHrVIUg8c3BdIUg5ISVIOEH00QZL6gBPoKpI0UFLiypGk4rh51CJJPXBwXyBJOSAlSTlA9NEESeoDTqCrSNJASYkrR5KK4+ZRiyT1wMF9gSTlgJQk5QDRRxMkqQ84ga4iSQMlJa4cSSqOm0ctktQDB/cFRSXdunUrnE7nY/d0OBx48sknH7u8XH117Ngx7uIJaVBRSVnS5eXl9PwJMXD1mRCpeJcNiqS8k6D25CNAksrHllrmRIAk5QSSmpGPAEkqH1tqmRMBkpQTSGpGPgIkqXxsqWVOBEhSTiCpGfkIkKTysaWWORC4desWzLqeGJ+azKE18U3QcVLx7B7rmkePHkXS0Ag0LH0ay6b2witZE3Djxo2g5EySBgW7ejf64MEDVFWUYPaLkbj2+97AqX4dzyNlIUgcHI6mpibFg5dV0i+//BJnz57Fjh07ULl8OSwD+iM5IR7rKyqwZ88eXLhwAd9//73iSdMGvRP46KOPkDk5CdXzftUpp0tS9nrjjd54ZXIkKssLFe03WSQ9fvw45makI9lmQV6MHRtNOrxpDscBawQarBHYYQrDWqsec2IcGGY0oGThQrz//vveydGnihDYu3cvUhIjcHptT6+Custak9sb6RMTcfnyZUVi4yrpu+++i/QxozE/2oaDtkj8bViU3+f1eDvqTWGYGm1HXnYW2traFEmcNvKIANvbLS2YicUZOtze730EdRfU9b5lXU+kOiOwa9cu2VFyk/S1dVV4yW7F4QDl9CbwdmMYnEYDGhsbZU+cNgCwQWX8r6Pw5qJf+h09XXK6v955qw+KMsOxJC8LX3zxhWxIJUv6ww8/oHDWTCyxGXEzweF35PQmp/tn52PNmGa3oLa6WrakqWGgtuY1TBkbiUube4kS1F3W3YVPI/U5e8fvDznYSpa0YEYWyk06yXK6i3oj3oEcuxm1r78uR86abvPmzZuYmz0RFTMH4F9NfSUL6pL1ck0vZKREoqZ6A3e+kiStKluBpVY9V0FdsrLvqpl2Cw4ePMg9aa022NzcDOcQHQ4vD+Emp0tS9nr/aF9UzuqPOdPHcz2mKlpS9gs+02GVRVCXqH8eZEaiIZJrwloUtL29HWtWFSNnUiTadvzv2Ke7YDzfN5WFYMTgcBw5coQLblGSsqRT44bgVJRBVkmZrFsNoVg0I5tLslps5MMPP8TUSU5snh/4L3cewv61nh1TjcDqssWSj6mKknT37t0ocZhlF9Q1ok6xWfDBBx9o0TFJObN+en5EBM5U+T/2yUNMb238LvcZpE8cjqtXr4rORZSkk0c60RJjUkzSOmMYVuTniU5SixUvXryIeFsP/L1B2RHUm6hvlYSgNPcF0d0gWNKPP/4YmTHSDzW5RslAXq/G2ZBoMoAd7qJH4ATGjXKoQtL9xU9jz5tbAg+8S0nBkrIZhrU2+b+LdpU3225RbBquC6Of7GLV6hI0LpPnl7y3EbO7z+ZOjsCVK1dEcxQsaWl+HhqsfI+LdhXS2/JKkw4HDhwQnagWK7a2tiJvSoQsh5u6E7Lr52yqlc1qSXkIljR74gs4G2NU7PuoS9gthlBUr18vJVdN1h0zwoJ7b/cJmqhvl4ZgXeUySewFS/rSyCRcHGxRXNJd5nCsXlosKVktVmaHgNhxy64jnFLLC6focP78eUnoBUua5hyBvwRB0t3mcFQUFUlKVouV2c3U2BlOSknpvp27B/pgrNMiGbtgSX+bmoJzCh5+cu3utxlDsWnNGskJa60BdkTkuQQ9/nlI+V3+kRUhqCxbLBm5YEmX5MxGoy1C8d19pSEM7MRceggnUFaSi3dW9VB8NF2Urus4HVB4xJ41BEtat20bNkT0V1zSOXZzx+UmnuHTUiAETp8+jaXTwhWV9B+H+mDUMAN+/PHHQEL0WUawpGwmY4Zd2R9OtxIcHSeafPfddz6ToZXeCbDryNgJH/9u5ndqnvt3T2/v2chdXrLAe0ACPxUsKWt/TEy0or/w91p0dJKJwI7tWrxk8Wz8YY1yu/zizDCcOXOmaxiilkVJys6aX20IV2yXz0bulpYWUQlSpUcETpw4geXTQxXZ5bOTqdl5q/fv3+eCX5Sk7OKt4QY9LsTKfybUPosOMydO4JKslhv59ttvMXJoOB68I/8u/1RlD5QW5nDDLUpStnX2S3u+Xd4zoa7E2TDOYsKlS5e4JazlhpYsnA52lae375A8P2Mj9smTJ7mhFi0pi2DxzBlYpx8o224/12bC9m1buSWr9YbYrXNWzRgoq6Ttx/oiaUg4eP7IlSTpN998g8yxY7BZBlGLLQaULSrQuldc8//qq68wepgOD08+unUOz9HT1dbZqp4oysviGrckSVkkt2/fxrSUsajSh3IZUdmVovk2E5bnLeSaKDX2iED+3HS8t1G+Xf7KGQPBRmyeD8mSsmDYiFqYMxvzHBa0Svgx1WSPRJrdgs2bNvHMkdpyI3Do0CGsnd1fll3+f072w+gEHb7++mu3LUp/y0VSVxgNDQ1wWsxYZTOA3eTBNe/u7/VElAEFUVawM6zYCRH0kI/AnTt3Oq57cu2eeb7+aWNPFMzL4B48V0lZdOx7z/bt2zEuPg7TYxyosuqx36LDH6ONYJcovzfIBCblTnM4yh1mpEXZMTU1hW6tw71ru29wzvQJqJz+CzSWhnB9po/qgcOHD3e/YZFruEvqHge7nHbnzp1YVpCP2WmTYe7dC8MMesxNfxkVy0o7zrS/fv26exV6rwABduXt5ur13J/Vm6rAjqHzfsgqaddg1XD/9a4x0bL6CZCk6u8jzUdIkmpeAfUDIEnV30eaj5Ak1bwC6gdAkqq/jzQfIUmqeQXUD4AkVX8faT5CklTzCqgfAEmq/j7SfIQkqeYVUD8AklT9faT5CElSzSugfgCKShobG4vc3Fz1U6EIVUVAMUnZX1T//JmfwRprVhUACkb9BBSTtHZbLXQv90aosy+dfa9+L1QVoSKSfvbZZzAOjUTibj0GVQ1EctooVUGgYNRNQBFJ84oXwJzfF881mTqejml67Nu3T91kKDrVEJBdUnYrakfKIzldksbXRcCaYMK9e/dUA4ICUS8B2SWdMDUV0RUDOkdRl6jmec9i5doy9ZKhyFRDQFZJ2T8sR6Ub/k9QJmpSoxHWkQZ88sknqoFBgaiTgGySsru4xSQ5EFer8yopE9Ve+iymvcL/Om11oqaoxBKQTdINr62HLWdgt4K6dvtRk0w4deqU2PipngYIyCJpW1sbLMP1cDZ439W7BGWvsRvD4Bw/XAOoKUWxBGSRNCd/JqxL+vkdRV2yOrJ0qK+vF5sD1XvMCXCXlN3LKWqC5yEnl4zdvQ6rj4R+kA6ff/75Y46b0hNDgLukbDaJzSp1J2R3n5sW9EFxWaGYHKjOY06Aq6RsFsmRqRcsqEtce7IR7D5F9CAC7gS4SXr37l1Y401IqIsQLamjrD/Ssie5x0fviQC4SbpybTnYLJJrVBT7GvUbI5qbm6lriEAnAS6Sslkj60gjkg4aJUs65PVwxCUPRnt7e2eQ9EbbBLhIymaN2OyR2NGzaz37rDDUbKnRds9Q9p0EJEvKZovYrFFX0aQsJ+7RwxQfiU8//bQzUHqjXQKSJH348CGc44Yj9tUwrpIywc0FfbGgaL52e4Yy7yQgSdL6+jfgyBL/a97faOt43oTW1tbOYOmNNgmIlpTNDuljdWCzRf5kE7s+evUAjM9I0WbPUNadBERLWrSiEGyWSKyAgdaLyjDQP5N0dpc234iSlM0KsdmhQEWTUi5uiw7RTnvHH5pps4soa1GSpmW/iKiy/opIygS3zhmIda9WUW9plIBgSdlsEJsVkjI6Cq3Lzku1JOpx7do1jXaTttMWJCmbBRo6OhZsVkioaFLLW4v6YdbCbG33lkazFyRpTe1msNkgqcKJrc/OU21padFoV2k3bUGS6gfrYBkZCXuyKShP3ZBQpGaM1W5vaTRzQZJqlBGlHWQCJGmQO4A2758ASeqfEZUIMgGSNMgdQJv3T4Ak9c+ISgSZAEka5A6gzfsnQJL6Z0QlgkyAJA1yB9Dm/RMgSf0zohJBJkCSBrkDaPP+CZCk/hlRiSATIEmD3AG0ef8ESFL/jKhEkAn8F8z+ZaER3TCEAAAAAElFTkSuQmCC[/img][/td][td][img]data:image/png;base64,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[/img][/td][/tr][/table]
Diego is solving this system of equations:
[size=150][math]\begin{cases} \begin {align}4x + 3y &= 10\\ \text-4x + 5y &= 6 \end{align} \end{cases}[/math][br][br]Here is his work: [br][table][tr][td][math]\begin {align}4x + 3y &= 10\\ \text-4x + 5y &= \hspace{2mm}6 \quad+\\ \overline {\quad 0 + 8y} &\overline{ \hspace{1mm}= 16 \qquad}\\ y &= 2 \end{align}[/math][/td][td][math]\begin {align}4x + 3(2) &= 10\\ 4x + 6 &= 10\\ 4x &= 4\\ x &= 1 \end{align}[/math][br][/td][/tr][/table][/size][br][size=150]Make sense of Diego’s work and discuss with a partner:[/size][br][list][*]What did Diego do to solve the system?[/*][*]Is the pair of [math]x[/math] and [math]y[/math] values that Diego found actually a solution to the system? How do you know?[/*][/list]
[size=150]Does Diego’s method work for solving these systems? Be prepared to explain or show your reasoning.[br][/size][math]\begin {cases} \begin {align}2x + y &= 4\\ x - y &= 11 \end {align} \end {cases}[/math]
[math]\begin {cases} \begin{align} 8x + 11y &= 37\\ 8x + \hspace{4.5mm} y &= \hspace{2mm} 7 \end{align} \end{cases}[/math]
Here are three systems of equations you saw earlier.
[table][tr][td]System A[/td][td]System B[/td][td]System C[/td][/tr][tr][td][math]\begin {cases}\begin {align}4x + 3y &= 10\\ \text-4x + 5y &= \hspace{2mm}6 \end{align} \end{cases}[/math][br][/td][td][math]\begin {cases} \begin {align}2x + y &= 4\\ x - y &= 11 \end {align} \end {cases}[/math][/td][td][math]\begin {cases} \begin{align} 8x + 11y &= 37\\ 8x + \hspace{4mm} y &= \hspace{2mm} 7 \end{align} \end{cases}[/math][/td][/tr][/table][size=150]For each system:[br][/size][list][*]Use graphing technology to graph the original two equations in the system. Then, identify the coordinates of the solution.[/*][*]Find the sum or difference of the two original equations that would enable the system to be solved. [/*][*]Graph the third equation on the same coordinate plane. Make an observation about the graph.[/*][/list]
Mai wonders what would happen if we multiply equations.
[size=150]That is, we multiply the expressions on the left side of the two equations and set them equal to the expressions on the right side of the two equations.[/size][br][br]In system B write out an equation that you would get if you multiply the two equations in this manner.
Does your original solution still work in this new equation?[br]
Use graphing technology to graph this new equation on the same coordinate plane.
Why is this approach not particularly helpful?
IM Alg1.2.14 Practice: Solving Systems by Elimination (Part 1)
Which equation is the result of adding these two equations?
[math]\begin{cases} \text-2x+4y=17 \\ 3x-10y = \text-3 \end{cases}[/math]
Which equation is the result of subtracting the second equation from the first?
[math]\begin{cases} 4x-6y=13 \\ \text-5x+2y= 5 \end{cases}[/math]
Solve this system of equations without graphing:
[math]\begin{cases} 5x+2y=29 \\ 5x - 2y= 41 \\ \end{cases}[/math]
Here is a system of linear equations:
[math]\begin{cases} 6x+21y=103 \\ \text-6x+23y=51 \\ \end{cases}[/math][br][br]Would you rather use subtraction or addition to solve the system? Explain your reasoning.
[size=150]Kiran sells [math]f[/math] full boxes and [math]h[/math] half-boxes of fruit to raise money for a band trip. He earns $5 for each full box and $2 for each half-box of fruit he sells and earns a total of $100 toward the cost of his band trip. The equation [math]5f+2h=100[/math] describes this relationship.[br][/size][br]Solve the equation for [math]f[/math].
Match each equation with the corresponding equation solved for a.
[size=150]The volume of a cylinder is represented by the formula [math]V=\pi r^2h[/math].[/size]
Find each missing height and show your reasoning.
Match each equation with the slope m and y-intercept of its graph.
Solve each system of equations.
[math]\begin{cases} 2x+3y=4 \\ 2x = 7y + 24\\ \end{cases}[/math]
[math]\begin{cases} 5x + 3y= 23 \\ 3y = 15x - 21 \\ \end{cases}[/math]
Elena and Kiran are playing a board game. After one round, Elena says, "You earned so many more points than I did. If you earned 5 more points, your score would be twice mine!"
[size=150]Kiran says, "Oh, I don't think I did that much better. I only scored 9 points higher than you did."[/size][br][br]Write a system of equations to represent each student's comment. Be sure to specify what your variables represent.[br]
If both students were correct, how many points did each student score? [br]