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Chapter 13: Problem 34

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} 3 x^{2}-2 y^{2}=1 \\ 4 x-y=3 \end{array}\right.$$

Short Answer

Expert verified
So, the solutions to the system of equations are \((1, 1)\) and \((19/29, -23/29)\).

Step by step solution

01

Solve the linear equation for one variable

Looking at the linear equation \(4x - y = 3\), it can be rearranged to solve for \(y\), which gives: \(y = 4x - 3\)
02

Substitution

Substitute \(y\) from the linear equation into the nonlinear equation, yielding: \(3x^{2} - 2(4x - 3)^{2} = 1\)
03

Solve the resultant equation

Now we should simplify and solve the equation for \(x\): \(3x^2 - 2(16x^2 -24x + 9) = 1\). Simplifying it, you get \(3x^2 - 32x^2 +48x -18 = 1\), which can be regrouped as \(-29x^2 +48x -19 = 0\). This can be solved by the quadratic formula \(x = [-b \pm \sqrt{(b^2 -4ac)}] / 2a\), which leads to \(x = 1\) or \(x = 19/29\)
04

Solve for the other variable

Substitute the derived x-values into the linear equation \(y = 4x - 3\) to find the corresponding y-values: For \(x = 1\), \(y = 4(1) - 3 = 1\), and for \(x = 19/29\), \(y = 4(19/29) - 3 = -23/29\).

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Most popular questions from this chapter

Use the vertex and intercepts to sketch the graph of each equation. If needed, find additional points on the parabola by choosing values of y on each side of the axis of symmetry. $$x=-y^{2}-6 y+7$$

Use the vertex and intercepts to sketch the graph of each equation. If needed, find additional points on the parabola by choosing values of y on each side of the axis of symmetry. $$x=-2(y+6)^{2}+2$$

The equation of a parabola is given. Determine: a. if the parabola is horizontal or vertical. b. the way the parabola opens. c. the vertex. $$y=x^{2}-4 x-1$$

Use the vertex and intercepts to sketch the graph of each equation. If needed, find additional points on the parabola by choosing values of y on each side of the axis of symmetry. $$x=-2 y^{2}-4 y$$

(GRAPH CANT COPY) Find the coordinates of the vertex for the horizontal parabola defined by the given equation. $$x=2(y-6)^{2}$$
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