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XYZ - Solve Systems of Equations

Teacher Notes

Students are presented with three equations with three unknowns. All equations have more than one variable. In all cases, all variables in one equation are in a second equation. The second equation contains at least one other variable. Students learn to replace the set of variables with its value in order to find the value of the other variable in the second equation. Once the value of one variable is known, its value can be used to figure out the values of the other variables. Encourage students to check solutions by replacing variables with their values.

Solutions

  1. x = 4, \ y = 7, \ z = 6
  2. x = 6, \ y = 8, \ z = 9
  3. x = 3, \ y = 6, \ z = 5
  4. x = 5, \ y = 4, \ z = 7
  5. x = 9, \ y = 5, \ z = 8
  6. x = 8, \ y = 10, \ z = 7
  7. x = 12, \ y = 9, \ z = 11
  8. x = 7, \ y = 5, \ z = 9

Eric wrote these equations to represent pictures of scales with blocks. Figure out the value of each unknown.

A: x + y = 12 \qquad \qquad \qquad x = \underline{\;\;\;\;\;\;\;\;\;\;}\\  B: x + x + y = 19 \ \qquad \qquad y = \underline{\;\;\;\;\;\;\;\;\;\;}\\C: x + z = 10 \qquad \qquad \qquad z = \underline{\;\;\;\;\;\;\;\;\;\;}

& \mathbf{Describe} && \text{There are three equations. There are 3 unknowns.}\\&&& \text{All equations have more than one unknown.}\\& \mathbf{My \ Job} && \text{Figure out the values of} \ x, y, \ \text{and} \ z.\\& \mathbf{Plan} && \text{All equations have more than one unknown.}\\&&& \text{Equations} \ A \ \text{and} \ B \ \text{are related. In Equation} \ A, x + y = 12\\&&& \text{That same} \ x + y \ \text{can be replaced by its value} \ 12 \ \text{in Equation} \ B.\\&&& \text{Replace} \ x + y \ \text{with} \ 12. \ \text{Figure out the value of the extra} \ x.\\&&& \text{Replace all} \ x’s \ \text{with that value in all equations and continue to solve for the}\\&&& \text{other unknowns.}\\& \mathbf{Solve} && A: x + y = 12.\\&&& B: \text{Replace} \ x + y \ \text{with} \ 12. \ x + 12 = 19.\\&&& \text{So,} \ x = 19 - 12, \ \text{or} \ 7.\\&&& \text{Replace all} \ x’s \ \text{with} \ 7.\\&&& C:  7 + z = 10, \ \text{so} \ z = 10 - 7, \ \text{or} \ 3.\\&&& A: 7 + y = 12, \ \text{so} \ y = 12 - 7, \ \text{or} \ 5.\\ &&& \text{So} \ x = 7, \ y = 5, \ z = 3\\& \mathbf{Check} && \text{Replace each variable with its value.}\\&&& A: 7 + 5 = 12; \ B: 7 + 7 + 5 = 19; \ C: 7 + 3 = 10.

Eric wrote these equations to represent pictures of scales with blocks. Figure out the value of each unknown.

1. \quad y + z + z = 19 \qquad \qquad \qquad x = \underline{\;\;\;\;\;\;\;\;\;\;}\!\\{\;} \quad \ y + z = 13 \qquad \qquad \qquad \qquad y = \underline{\;\;\;\;\;\;\;\;\;\;}\!\\{\;} \quad \ x + y + y = 18 \qquad \qquad \qquad z = \underline{\;\;\;\;\;\;\;\;\;\;}

2. \quad x + x + y = 20 \qquad \qquad \qquad x = \underline{\;\;\;\;\;\;\;\;\;\;}\!\\{\;} \quad \ y + z + z = 26 \qquad \qquad \qquad \ y = \underline{\;\;\;\;\;\;\;\;\;\;}\!\\{\;} \quad \ y + z = 17 \qquad \qquad \qquad \qquad z = \underline{\;\;\;\;\;\;\;\;\;\;}

3. \quad x + z = 8 \qquad \qquad \qquad \qquad x = \underline{\;\;\;\;\;\;\;\;\;\;}\!\\   {\;} \quad \ x + z + z = 13 \qquad \qquad \qquad y = \underline{\;\;\;\;\;\;\;\;\;\;}\!\\{\;} \quad \ x + y + z = 14 \qquad \qquad \qquad z = \underline{\;\;\;\;\;\;\;\;\;\;}

4. \quad x + z + z = 19 \qquad \qquad \qquad x = \underline{\;\;\;\;\;\;\;\;\;\;}\!\\{\;} \quad \ y + y + z = 15 \qquad \qquad \qquad y = \underline{\;\;\;\;\;\;\;\;\;\;}\!\\{\;} \quad \ x + z = 12 \qquad \qquad \qquad \quad \ \ z = \underline{\;\;\;\;\;\;\;\;\;\;}

Eric wrote these equations to represent pictures of scales with blocks. Figure out the value of each unknown.

5. \quad x + y + z = 22 \qquad \qquad \qquad x = \underline{\;\;\;\;\;\;\;\;\;\;}\!\\{\;} \quad \ x + z = 17 \qquad \qquad \qquad \quad \ \ y = \underline{\;\;\;\;\;\;\;\;\;\;}\!\\{\;} \quad \ y + z + z = 21 \qquad \qquad \qquad z = \underline{\;\;\;\;\;\;\;\;\;\;}

6. \quad x + y + z = 25 \qquad \qquad \qquad x = \underline{\;\;\;\;\;\;\;\;\;\;}\!\\{\;} \quad \ y + z = 17 \qquad \qquad \qquad \quad \ \ y = \underline{\;\;\;\;\;\;\;\;\;\;}\!\\{\;} \quad \ x + x + z = 23 \qquad \qquad \qquad z = \underline{\;\;\;\;\;\;\;\;\;\;}

7. \quad x + x + y = 33 \qquad \qquad \qquad x = \underline{\;\;\;\;\;\;\;\;\;\;}\!\\{\;} \quad \ x + z + z = 34 \qquad \qquad \qquad \ y = \underline{\;\;\;\;\;\;\;\;\;\;}\!\\{\;} \quad \ x + y = 21 \qquad \qquad \qquad \quad \ \ z = \underline{\;\;\;\;\;\;\;\;\;\;}

8. \quad y + y + z + z = 28 \qquad \qquad x = \underline{\;\;\;\;\;\;\;\;\;\;}\!\\{\;} \quad \ x + x + y = 19 \qquad \qquad \quad \ \ y = \underline{\;\;\;\;\;\;\;\;\;\;}\!\\{\;} \quad \ x + x + y + y = 24 \qquad \qquad z = \underline{\;\;\;\;\;\;\;\;\;\;}

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