Solve for X, Y, And Z. X + 2y + z = 3 2x - y + 2z = 6 3x + y - z = 5 -----------

Question

Solve for X, Y, And Z.
X + 2y + z = 3
2x - y + 2z = 6 3x + y - z = 5
---------------- Solve for X, Y, And Z.
X/2 - 3Y/4 - Z/7 = 8
X/2 + Y/4 - Z/5 = 0
X/4 - Z + Y/8 = 0 ---------------- The Parabola Y= ax2 + bx + c passes through the points (1,2), (2,4) and (3,8). Determine the Values of A, B and C.
2x - y + 2z = 6 3x + y - z = 5
---------------- Solve for X, Y, And Z.
X/2 - 3Y/4 - Z/7 = 8
X/2 + Y/4 - Z/5 = 0
X/4 - Z + Y/8 = 0 ---------------- The Parabola Y= ax2 + bx + c passes through the points (1,2), (2,4) and (3,8). Determine the Values of A, B and C.

Explanation / Answer

The first problem can be answered by using both the combination and substitution methods. 1. Add the last two equations together to get 5x + z = 11. The y-values cancel out. Solving this for z you get z = 11 - 5x. 2. Substitute 11 - 5x in for the z-value in the first equation: x + 2y + 11 - 5x = 3. Solving this for y you get y = 2x - 4. 3. Add the first and third equations to get 4x + 3y = 8. 4. Substitute the equation for y from step (2) into the equation from step (3). This leaves you with 4x + 3(2x - 4) = 8. Solve this for x, then plug that x-value into the equation from step (2) to solve for y. Finally, plug in both x and y values into any of the given equations to solve for z. For the second problem, you want to use substitution but first you want to get rid of the fractions to make the arithmetic easier. 1. Multiply every term in each equation by the lowest common denominator. Equation 1, LCD = 28: 28(x/2 - 3y/4 - z/7) = 28(8). Equation 2, LCD = 20: 20(x/2 + y/4 - z/5) = 20(0). Equation 3, LCD = 8: 8(x/4 - z + y/8) = 8(0). This leaves you with Equation 1: 14x - 21y - 4z = 224, Equation 2: 10x + 5y - 4z = 0, Equation 3: 2x - 8z+ y = 0. 2. Take the third equation (2x - 8z + y = 0) and solve for y, leaving you with y = 8z - 2x. 3. Substitute this y-value into equation 2: 10x + 5(8z - 2x) - 4z = 0. When you simplify this, you find that 10x + 40z - 10x - 4z = 0. The x terms cancel out, so 36z = 0. Therefore, z = 0. 4. From here, you can substitute the equation for y obtained in step (2) and z = 0 into the first given equation (14x - 21y - 4z = 224) and solve for x. Plug z = 0 and the x-value you obtain into the equation from step (2) to solve for y. For the last problem, you have to plug in the points for x and y in the parabolic equation. This gives you 3 equations: 2 = a + b + c, 4 = 4a + 2b + c, and 8 = 9a + 3b + c. 1. Solve for the a-term in equation 1: a = 2 - b - c 2. Substitute this into equation 2: 4 = 4(2 - b - c) + 2b + c. Simplifying: 4 = 8 - 4b - 4c + 2b + c. Simplifying again gives you: -4 = -2b - 3c. 3. Use this equation to solve for b: -2b = 3c - 4. Dividing both sides by -2 gives you b = (3/-2)c + 2. 4. Substitute the equation for a from step (1) into equation 3: 8 = 9(2 - b - c) + 3b + c. Simplifying: 8 = 18 - 9b - 9c + 3b + c. Simplifying again: -10 = -6b - 8c. 5. Substitute the equation for b from step (3) into the equation from step (4): -10 = -6[(3/-2)c + 2] - 8c. Simplifying: -10 = 9c - 12 - 8c. Simplifyig again: 2 = c. 6. Plug this in for the equaiton from step (3) to solve for b. Then plug both the b and c values into any of the original equations to solve for a.

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