{:[-x-5y+z=17],[-5x-5y+5z=5],[2x+5y-3z=-10]:}

Question

Bytelearn · Accepted Answer

Elimination Method: First, let's solve the system of equations using the elimination method. We have the system: 1) -x - 5y + z = 17 2) -5x - 5y + 5z = 5 3) 2x + 5y - 3z = -10 Let's start by eliminating y from equations 1 and 3 by adding them together.Equation 4: Adding equation 1 and equation 3: (-x - 5y + z) + (2x + 5y - 3z) = 17 + (-10) This simplifies to: -x + 2x - 5y + 5y + z - 3z = 7 Which further simplifies to: x - 2z = 7 Let's call this equation 4.Equation 5: Now, let's eliminate y from equations 2 and 3 by multiplying equation 3 by -1 and adding it to equation 2. Multiplying equation 3 by -1 gives us: -2x - 5y + 3z = 10 Let's call this equation 5.Equation 6: Adding equation 2 and equation 5: (-5x - 5y + 5z) + (-2x - 5y + 3z) = 5 + 10 This simplifies to: -5x - 2x - 5y - 5y + 5z + 3z = 15 Which further simplifies to: -7x + 8z = 15 Let's call this equation 6.Equation 7: Now we have two equations with two variables (equations 4 and 6): 4) x - 2z = 7 6) -7x + 8z = 15 Let's multiply equation 4 by 7 to help eliminate x: 7(x - 2z) = 7(7) 7x - 14z = 49 Let's call this equation 7.Value of z: Adding equation 6 and equation 7: (-7x + 8z) + (7x - 14z) = 15 + 49 This simplifies to: -7x + 7x + 8z - 14z = 64 Which further simplifies to: -6z = 64 Dividing both sides by -6 gives us: z = -64/6 z = -32/3Substitute for x: Now that we have the value of z, we can substitute it back into equation 4 to find x: x - 2(-32/3) = 7 x + 64/3 = 7 To solve for x, we need to subtract 64/3 from both sides: x = 7 - 64/3 x = (21 - 64)/3 x = -43/3Find y using Equation 1: Now we have the values for x and z. We can substitute these into any of the original equations to find y. Let's use equation 1: -x - 5y + z = 17 -(-43/3) - 5y + (-32/3) = 17 43/3 - 5y - 32/3 = 17 Combining like terms gives us: 11/3 - 5y = 17 To solve for y, we need to subtract 11/3 from both sides and then divide by -5: -5y = 17 - 11/3 -5y = (51 - 11)/3 -5y = 40/3 y = -40/3 / -5 y = 40/3 * 1/5 y = 8/3Substitution Method: Now let's solve the system using the substitution method. We will start by expressing x from equation 1 in terms of y and z: -x - 5y + z = 17 x = -5y + z - 17 Let's call this equation 8.Incorrect Substitution: Next, we substitute equation 8 into equation 2: -5(-5y + z - 17) - 5y + 5z = 5 This simplifies to: 25y - 5z + 85 - 5y + 5z = 5 Which further simplifies to: 20y + 85 = 5 Subtracting 85 from both sides gives us: 20y = -80 Dividing both sides by 20 gives us: y = -4 However, this is a math error because we did not substitute correctly. We should have substituted for x in all terms of equation 2, not just the first term.

$\begin{array}{l} -x-5 y+z=17 \\ -5 x-5 y+5 z=5 \\ 2 x+5 y-3 z=-10 \end{array}$

Full solution

More problems from Write an equation word problem

Related topics

−x−5y+z=17−5x−5y+5z=52x+5y−3z=−10 \begin{array}{l} -x-5 y+z=17 \\ -5 x-5 y+5 z=5 \\ 2 x+5 y-3 z=-10 \end{array} −x−5y+z=17−5x−5y+5z=52x+5y−3z=−10​

$\begin{array}{l} -x-5 y+z=17 \\ -5 x-5 y+5 z=5 \\ 2 x+5 y-3 z=-10 \end{array}$