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We want to solve the following system of equations.

{[y=4x(x-2)],[x+y=2]:}
One of the solutions to this system is
(2,0).
Find the other solution.
Your answer must be exact.

We want to solve the following system of equations. $\newline$ $\left\{\begin{array}{l} y=4 x(x-2) \\ x+y=2 \end{array}\right.$ $\newline$ One of the solutions to this system is $(2,0)$ . $\newline$ Find the other solution. $\newline$ Your answer must be exact.

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Find what percent one number is of another: word problems

Full solution

Q. We want to solve the following system of equations.

\newline

\left\{\begin{array}{l} y=4 x(x-2) \\ x+y=2 \end{array}\right.

\newline

One of the solutions to this system is

(2,0)

.

\newline

Find the other solution.

\newline

Your answer must be exact.

Substitute and simplify: Substitute the expression for $y$ from the first equation into the second equation. $\newline$ We have $y = 4x(x - 2)$ and $x + y = 2$ . Substituting $y$ in the second equation gives us $x + 4x(x - 2) = 2$ .
Expand and combine like terms: Expand the equation $x + 4x(x - 2) = 2$ . $\newline$ This gives us $x + 4x^2 - 8x = 2$ .
Factor the quadratic equation: Simplify the equation by combining like terms. $\newline$ This results in $4x^2 - 7x + 2 = 0$ .
Solve for x: Factor the quadratic equation $4x^2 - 7x + 2 = 0$ . $\newline$ The factors of this quadratic equation are $(4x - 1)(x - 2) = 0$ .
Substitute and simplify: Solve for $x$ by setting each factor equal to zero. $\newline$ Setting $4x - 1 = 0$ gives us $x = \frac{1}{4}$ . $\newline$ Setting $x - 2 = 0$ gives us $x = 2$ , but we already know that $(2,0)$ is one solution, so we are looking for the other solution.
Check the solution: Substitute $x = \frac{1}{4}$ into the first equation to find the corresponding value of $y$ . $\newline$ Substituting into $y = 4x(x - 2)$ gives us $y = 4 \cdot \left(\frac{1}{4}\right) \cdot \left(\left(\frac{1}{4}\right) - 2\right)$ .
Check the solution: Substitute $x = \frac{1}{4}$ into the first equation to find the corresponding value of $y$ . Substituting into $y = 4x(x - 2)$ gives us $y = 4 \cdot \left(\frac{1}{4}\right) \cdot \left(\left(\frac{1}{4}\right) - 2\right)$ . Simplify the expression to find the value of $y$ . This gives us $y = 4 \cdot \left(\frac{1}{4}\right) \cdot \left(-\frac{7}{4}\right)$ , which simplifies to $y = -7$ .
Check the solution: Substitute $x = \frac{1}{4}$ into the first equation to find the corresponding value of $y$ . $\newline$ Substituting into $y = 4x(x - 2)$ gives us $y = 4 \cdot \left(\frac{1}{4}\right) \cdot \left(\left(\frac{1}{4}\right) - 2\right)$ . Simplify the expression to find the value of $y$ . $\newline$ This gives us $y = 4 \cdot \left(\frac{1}{4}\right) \cdot \left(-\frac{7}{4}\right)$ , which simplifies to $y = -7$ . Check the solution $\left(\frac{1}{4}, -7\right)$ in the second equation $x + y = 2$ . $\newline$ Substituting $x = \frac{1}{4}$ and $y = -7$ into the equation gives us $y$ $1$ , which simplifies to $y$ $2$ . There is a math error in the previous steps.

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