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Problem 10

# Find all solutions of the given system of equations and check your answer graphically. \begin{aligned}-\frac{2 x}{3}+\frac{y}{2} &=-\frac{1}{6} \\ \frac{x}{4}-y &=-\frac{3}{4} \end{aligned}

Expert verified
The solution of the given system of equations is $$(5, \frac{5}{3})$$. To check this answer graphically, graph the lines with equations $$y = \frac{1}{3}x + \frac{1}{3}$$ and $$y = \frac{1}{4}x + \frac{3}{4}$$ and ensure that they intersect at the point $$(5, \frac{5}{3})$$.
See the step by step solution

## Step 1: Suspend the denominators in each equation

Start by multiplying both sides of each equation by the least common multiple of the denominators to eliminate the fractions. In this case, the least common multiple of 3, 2, and 4 is 12. For the first equation, multiply both sides by 12: $$12(-\frac{2x}{3}+\frac{y}{2})=12(-\frac{1}{6})$$ For the second equation, multiply both sides by 12: $$12(\frac{x}{4}-y)=12(-\frac{3}{4})$$

## Step 2: Simplify the equations

Perform the multiplication for each term in both equations: First equation: $$-8x + 6y = -2$$ Second equation: $$3x - 12y = -9$$

## Step 3: Solve the equations

There are many ways to solve a system of equations, such as substitution, elimination, and matrices. In this case, we will use the substitution method. Solve the first equation for y: $$-8x + 6y = -2 => y = \frac{1}{3}x + \frac{1}{3}$$ Now, substitute this expression into the second equation: $$3x - 12(\frac{1}{3}x + \frac{1}{3}) = -9$$

## Step 4: Solve for x

Now we just need to solve the above expression for x: $$3x - 4x - 4 = -9 => -x - 4 = -9$$ $$-x = -5 => x = 5$$

## Step 5: Solve for y

Use the value of x found in Step 4 and substitute it into the y expression derived in Step 3: $$y = \frac{1}{3}(5) + \frac{1}{3} => y = \frac{5}{3}$$

## Step 6: Write the solution as an ordered pair

Now, write down the solution as an ordered pair (x, y): $$(5, \frac{5}{3})$$

## Step 7: Rewrite the equations in slope-intercept form for graphing

To graph the lines, rewrite both original equations in slope-intercept form (y = mx + b): First equation: $$y = \frac{1}{3}x + \frac{1}{3}$$ Second equation: $$y = \frac{1}{4}x + \frac{3}{4}$$ The student should now graph these two lines and verify that they intersect at the point $$(5, \frac{5}{3})$$.

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