Simultaneous equations: elimination questions with answers

Questions

1. Question 1Same sign: subtract

   Solve the simultaneous equations $-6x + 4y = -46$ and $-3x + 4y = -19$.

   Show the worked solution

   1. Step 1: Number the equations$-6x + 4y = -46$ ... (1)$-3x + 4y = -19$ ... (2)
   2. Step 2: Eliminate y: add or subtract the equations?The $y$-terms have the same sign, so subtract.
   3. Step 3: Now the y-terms cancel. Solve for x$(2) - (1)$:$3x = 27$$x = \dfrac{27}{3}$$= 9$
   4. Step 4: Substitute your value of x into (1) to find y$-6(9) + 4y = -46$$4y = 8$$y = \dfrac{8}{4}$$= 2$Check in (2): $-3(9) + 4(2)$$= -19$, the right-hand side of (2).
2. Question 2Opposite signs: add

   Solve the simultaneous equations $-3x + 2y = -20$ and $7x - 2y = 36$.

   Show the worked solution

   1. Step 1: Number the equations$-3x + 2y = -20$ ... (1)$7x - 2y = 36$ ... (2)
   2. Step 2: Eliminate y: add or subtract the equations?The $y$-terms have opposite signs, so add.
   3. Step 3: Now the y-terms cancel. Solve for x$(1) + (2)$:$4x = 16$$x = \dfrac{16}{4}$$= 4$
   4. Step 4: Substitute your value of x into (1) to find y$-3(4) + 2y = -20$$2y = -8$$y = \dfrac{-8}{2}$$= -4$Check in (2): $7(4) - 2(-4)$$= 36$, the right-hand side of (2).
3. Question 3Multiply one equation first

   Solve the simultaneous equations $-2x + 9y = 60$ and $-3x + 3y = 27$.

   Show the worked solution

   1. Step 1: Number the equations$-2x + 9y = 60$ ... (1)$-3x + 3y = 27$ ... (2)
   2. Step 2: Neither add nor subtract gets rid of y yet: multiply so the y-terms are the same size$(2) \times 3$:$-9x + 9y = 81$ ... (3)
   3. Step 3: Eliminate y: add or subtract the equations?The $y$-terms have the same sign, so subtract.
   4. Step 4: Now the y-terms cancel. Solve for x$(1) - (3)$:$7x = -21$$x = \dfrac{-21}{7}$$= -3$
   5. Step 5: Substitute your value of x into (1) to find y$-2(-3) + 9y = 60$$9y = 54$$y = \dfrac{54}{9}$$= 6$Check in (2): $-3(-3) + 3(6)$$= 27$, the right-hand side of (2).
4. Question 4Multiply both equations first

   Solve the simultaneous equations $-3x + 4y = 13$ and $4x + y = -11$.

   Show the worked solution

   1. Step 1: Number the equations$-3x + 4y = 13$ ... (1)$4x + y = -11$ ... (2)
   2. Step 2: Neither add nor subtract gets rid of x yet: multiply so the x-terms are the same size$(1) \times 4$:$-12x + 16y = 52$ ... (3)$(2) \times 3$:$12x + 3y = -33$ ... (4)
   3. Step 3: Eliminate x: add or subtract the equations?The $x$-terms have opposite signs, so add.
   4. Step 4: Now the x-terms cancel. Solve for y$(3) + (4)$:$19y = 19$$y = \dfrac{19}{19}$$= 1$
   5. Step 5: Substitute your value of y into (1) to find x$-3x + 4(1) = 13$$-3x = 9$$x = \dfrac{9}{-3}$$= -3$Check in (2): $4(-3) + (1)$$= -11$, the right-hand side of (2).