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Find all solutions of the given system of equations, and check your answer graphically. $$ \begin{array}{l} 3 x-2 y=6 \\ 2 x-3 y=-6 \end{array} $$

Short Answer

Expert verified
The solution to the given system of equations is \(x = 6\) and \(y = 6\). This is verified graphically as the point (6,6) lies at the intersection of the two lines representing the equations.
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Step 1: Solve the system of equations algebraically

: We can use the elimination method. We begin by multiplying both equations by necessary multiples such that the coefficients of one variable in both equations are equal (either x or y). Then, we eliminate that variable by adding or subtracting the equations. In this case, we multiply the first equation by 2 and the second equation by 3. First equation: \(6x - 4y = 12\) Second equation: \(6x - 9y = -18\) Now, we subtract the second equation from the first equation to eliminate the variable \(x\). \((6x-4y)-(6x-9y) = 12 - (-18)\) We get the following equation: \(5y = 30\) Now, let's find the value of y by dividing both sides by 5: \(y = 6\) Now that we have the value of y, we plug it into either equation to solve for x. Let's use the first equation: \(3x - 2(6) = 6\) \(3x - 12 = 6\) Now, add 12 to both sides: \(3x = 18\) Divide by 3: \(x = 6\) Thus, the solution to the given system is: \(x = 6\) and \(y = 6\).

Step 2: Verify the solution graphically

: To verify the solution graphically, we plot the two given equations on the same coordinate plane. First, we rewrite the equations in slope-intercept form (y = mx + b): First equation: \(-2y = -3x + 6\) \(y = \frac{3}{2}x - 3\) Second equation: \(-3y = -2x - 6\) \(y = \frac{2}{3}x + 2\) Now, we plot both lines on the same coordinate plane. The first line has a slope of \(\frac{3}{2}\) and a y-intercept of -3. The second line has a slope of \(\frac{2}{3}\) and a y-intercept of 2. The point of intersection of both lines will represent our solution. According to our calculations, the point of intersection is (6,6). If this point lies on both lines, our solution is correct. From the graph, we can see that the point (6,6) does indeed lie at the intersection of the two lines, which confirms our solution. Therefore, the solution to the given system of equations is correct: \(x = 6\) and \(y = 6\).

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