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Chapter 20: Problem 586

Solve the system \(\mathrm{xy}=24\) \(y-2 x+2=0\)

Short Answer

Expert verified
The solution to the given system of equations is \((x, y) = (4, 6)\) and \((x, y) = (-3, -8)\).

Step by step solution

01

Solve the second equation for y

To solve the second equation for y, simply make y the subject of the equation: \(y - 2x + 2 = 0\) Adding \(2x - 2\) to both sides, we find: \(y = 2x - 2\)
02

Substitute the value of y into the first equation

Now that we have an expression for y, we'll use it to substitute in the first equation: \(\mathrm{xy} = 24\) Replace y with \((2x - 2)\): \(\mathrm{x(2x - 2)} = 24\)
03

Solve for x

Distribute x to the terms in the parentheses: \(\mathrm{2x^2 - 2x} = 24\) Subtract 24 from both sides to set the equation to zero: \(\mathrm{2x^2 - 2x - 24} = 0\) Divide the entire equation by 2 to simplify: \(\mathrm{x^2 - x - 12} = 0\) Factor the quadratic equation: \((x - 4)(x + 3) = 0\) Set each factor equal to zero and solve for x: \(x - 4 = 0\) --> \(x = 4\) \(x + 3 = 0\) --> \(x = -3\)
04

Solve for y

Use the expression of y from Step 1 and the values of x found in Step 3 to solve for the corresponding values of y: For \(x = 4\): \(y = 2(4) - 2\) \(y = 8 - 2\) \(y = 6\) For \(x = -3\): \(y = 2(-3) - 2\) \(y = -6 - 2\) \(y = -8\)
05

Write the solution

The solution to the given system of equations is \((x, y) = (4, 6)\) and \((x, y) = (-3, -8)\).

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Most popular questions from this chapter

Solve graphically \(\left\\{\begin{aligned} x^{2}+y^{2} &=13 \\ y &=x^{2}-1 \end{aligned}\right.\)

Solve for \(\mathrm{x}\) and \(\mathrm{y}\) : $$ \left\\{\begin{array}{l} 9 x^{2}-16 y^{2}=144 \\ x-2 y=4 \end{array}\right. $$

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Obtain the simultaneous solution set of the equations \(2 x^{2}+3 y^{2}=21\) \(3 x^{2}-4 y^{2}=23\)

Obtain the simultaneous solution set of the equations $$ \begin{aligned} y &=x^{2}-4 \\ 3 x^{2}+8 y^{2} &=75 \end{aligned} $$ by the graphical method.
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