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Problem 296

# Solve the equations $$3 \mathrm{x}+2 \mathrm{y}=1$$ and $5 \mathrm{x}-3 \mathrm{y}=8$ simultaneously.

Expert verified
The solution to the given system of equations $$3x + 2y = 1$$ and $$5x - 3y = 8$$ is the ordered pair $$(x, y) = (1, -1)$$.
See the step by step solution

## Step 1: Write the system of equations

First, let's write down the given system of equations. $$3x + 2y = 1$$ (1) $$5x - 3y = 8$$ (2)

## Step 2: Multiply each equation to make coefficients of y the same in absolute value

In order to eliminate one of the variables using the elimination method, we need the coefficients of y to be the same in both equations so that they cancel each other out when adding or subtracting the equations. To do so, we will multiply equation (1) by 3 and equation (2) by 2. Equation (1) * 3: $$9x + 6y = 3$$ (3) Equation (2) * 2: $$10x - 6y = 16$$ (4)

## Step 3: Add the new equations to eliminate y

Now that both new equations have the same coefficient for y, we can add them together to eliminate y. (3) + (4): $$9x + 6y + 10x - 6y = 3 + 16$$ $$19x = 19$$

## Step 4: Solve for x

Now we have a simple linear equation with only one variable. Solve for x: $$19x = 19 \Rightarrow x = \frac{19}{19} = 1$$

## Step 5: Substitute the value of x back into one of the original equations to find y

Now that we have the value for x, we can substitute it back into one of the original equations to find the value of y. Let's use equation (1) for this: $$3x + 2y = 1 \Rightarrow 3(1) + 2y = 1$$

## Step 6: Solve for y

Now we have a linear equation with only one variable, y. Solve for y: $$3 + 2y = 1 \Rightarrow 2y = -2 \Rightarrow y = \frac{-2}{2} = -1$$

## Step 7: Write the solution as an ordered pair

Now that we have the values of x and y, we can write the solution as an ordered pair: $$(x, y) = (1, -1)$$ So, the solution to the system of equations is (1, -1).

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