Find the values of x,y,z based on the equations: {:[x(z+1)=y^(2)],[(z+1)^(2)=xy],[x^(2)=y(z+1)]:}

Question

Bytelearn · Accepted Answer

Analyze System of Equations: Analyze the given system of equations to look for a strategy to solve for x, y, and z. We have a system of three equations with three variables. We can try to express one variable in terms of the others and substitute into the remaining equations to find a solution.Solve for y^2: Solve the first equation for y^2. From the first equation, we have y^2 = x(z+1).Substitute y^2 into Second Equation: Substitute y^2 from Step 2 into the second equation. We replace y^2 in the second equation (z+1)^2 = xy with x(z+1) to get (z+1)^2 = x^2(z+1).Simplify Equation: Simplify the equation from Step 3. We can divide both sides of the equation by (z+1), assuming z+1 is not equal to zero, to get z+1 = x^2.Substitute z+1 into Third Equation: Substitute z+1 from Step 4 into the third equation. We replace z+1 in the third equation x^2 = y(z+1) with x^2 to get x^2 = yx^2.Simplify Equation: Simplify the equation from Step 5. We can divide both sides of the equation by x^2, assuming x is not equal to zero, to get 1 = y.Substitute y=1 into First Equation: Substitute y=1 into the first equation to solve for x. We replace y with 1 in the first equation x(z+1) = y^2 to get x(z+1) = 1.Solve for x in terms of z: Solve for x in terms of z. We can divide both sides of the equation by (z+1), assuming z+1 is not equal to zero, to get x = 1/(z+1).Substitute y=1 and x into Second Equation: Substitute y=1 and x=1/(z+1) into the second equation to solve for z. We replace y with 1 and x with 1/(z+1) in the second equation (z+1)^2 = xy to get (z+1)^2 = (1/(z+1))(1).Simplify Equation: Simplify the equation from Step 9. We multiply both sides of the equation by (z+1) to get (z+1)^3 = 1.Solve for z: Solve for z. We take the cube root of both sides to get z+1 = 1, which gives us z = 0.Substitute z=0 into x equation: Substitute z=0 into the equation x=1/(z+1) to find x. We replace z with 0 in the equation x = 1/(z+1) to get x = 1/(0+1) = 1.Verify Solution: Verify the solution (x=1, y=1, z=0) in all three original equations. First equation: 1(0+1) = 1^2, which simplifies to 1 = 1. True. Second equation: (0+1)^2 = 1(1), which simplifies to 1 = 1. True. Third equation: 1^2 = 1(0+1), which simplifies to 1 = 1. True.

Find the values of $x,y,z$ based on the equations: $\newline$ $\ x(z+1)=y^{2}$ $\newline$ \((z+1)^{2}=xy\) $\newline$ $\ x^{2}=y(z+1)$

Full solution

More problems from Find derivatives of sine and cosine functions

Related topics

Find the values of x,y,zx,y,zx,y,z based on the equations: \newline x(z+1)=y2\ x(z+1)=y^{2} x(z+1)=y2\newline \((z+1)^{2}=xy\)\newline x2=y(z+1)\ x^{2}=y(z+1) x2=y(z+1)

Find the values of $x,y,z$ based on the equations: $\newline$ $\ x(z+1)=y^{2}$ $\newline$ \((z+1)^{2}=xy\) $\newline$ $\ x^{2}=y(z+1)$