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To solve the given system of equations:
[tex]\[ \begin{cases} 5x + 2y = 1 \\ -3x + 3y = 5 \end{cases} \][/tex]
we will use the method of solving simultaneous equations.
### Step 1: Isolate one of the variables in one of the equations
We start by isolating [tex]\(y\)[/tex] in the second equation. The second equation is:
[tex]\[ -3x + 3y = 5 \][/tex]
Divide the entire equation by 3 to simplify:
[tex]\[ -x + y = \frac{5}{3} \][/tex]
Rewriting it, we get:
[tex]\[ y = x + \frac{5}{3} \][/tex]
### Step 2: Substitute the expression for [tex]\(y\)[/tex] into the first equation
Next, we substitute [tex]\(y = x + \frac{5}{3}\)[/tex] into the first equation:
[tex]\[ 5x + 2y = 1 \][/tex]
Substituting [tex]\(y\)[/tex]:
[tex]\[ 5x + 2\left(x + \frac{5}{3}\right) = 1 \][/tex]
### Step 3: Solve for [tex]\(x\)[/tex]
Now, we solve for [tex]\(x\)[/tex]:
[tex]\[ 5x + 2x + \frac{10}{3} = 1 \][/tex]
Combine like terms:
[tex]\[ 7x + \frac{10}{3} = 1 \][/tex]
Next, subtract [tex]\(\frac{10}{3}\)[/tex] from both sides:
[tex]\[ 7x = 1 - \frac{10}{3} \][/tex]
Convert 1 to a fraction with a denominator of 3:
[tex]\[ 7x = \frac{3}{3} - \frac{10}{3} \][/tex]
Simplify the right-hand side:
[tex]\[ 7x = \frac{3 - 10}{3} \][/tex]
[tex]\[ 7x = -\frac{7}{3} \][/tex]
Now, divide both sides by 7:
[tex]\[ x = -\frac{7}{3} \div 7 \][/tex]
[tex]\[ x = -\frac{7}{3} \times \frac{1}{7} \][/tex]
[tex]\[ x = -\frac{1}{3} \][/tex]
### Step 4: Substitute [tex]\(x\)[/tex] back into the expression for [tex]\(y\)[/tex]
Now that we have [tex]\(x = -\frac{1}{3}\)[/tex], we substitute this value back into the expression for [tex]\(y\)[/tex]:
[tex]\[ y = x + \frac{5}{3} \][/tex]
[tex]\[ y = -\frac{1}{3} + \frac{5}{3} \][/tex]
Simplify the fraction:
[tex]\[ y = \frac{-1 + 5}{3} \][/tex]
[tex]\[ y = \frac{4}{3} \][/tex]
### Conclusion
The solution to the system of equations is:
[tex]\[ x = -\frac{1}{3}, \quad y = \frac{4}{3} \][/tex]
So the solution is:
[tex]\[ \left( x, y \right) = \left( -\frac{1}{3}, \frac{4}{3} \right) \][/tex]
[tex]\[ \begin{cases} 5x + 2y = 1 \\ -3x + 3y = 5 \end{cases} \][/tex]
we will use the method of solving simultaneous equations.
### Step 1: Isolate one of the variables in one of the equations
We start by isolating [tex]\(y\)[/tex] in the second equation. The second equation is:
[tex]\[ -3x + 3y = 5 \][/tex]
Divide the entire equation by 3 to simplify:
[tex]\[ -x + y = \frac{5}{3} \][/tex]
Rewriting it, we get:
[tex]\[ y = x + \frac{5}{3} \][/tex]
### Step 2: Substitute the expression for [tex]\(y\)[/tex] into the first equation
Next, we substitute [tex]\(y = x + \frac{5}{3}\)[/tex] into the first equation:
[tex]\[ 5x + 2y = 1 \][/tex]
Substituting [tex]\(y\)[/tex]:
[tex]\[ 5x + 2\left(x + \frac{5}{3}\right) = 1 \][/tex]
### Step 3: Solve for [tex]\(x\)[/tex]
Now, we solve for [tex]\(x\)[/tex]:
[tex]\[ 5x + 2x + \frac{10}{3} = 1 \][/tex]
Combine like terms:
[tex]\[ 7x + \frac{10}{3} = 1 \][/tex]
Next, subtract [tex]\(\frac{10}{3}\)[/tex] from both sides:
[tex]\[ 7x = 1 - \frac{10}{3} \][/tex]
Convert 1 to a fraction with a denominator of 3:
[tex]\[ 7x = \frac{3}{3} - \frac{10}{3} \][/tex]
Simplify the right-hand side:
[tex]\[ 7x = \frac{3 - 10}{3} \][/tex]
[tex]\[ 7x = -\frac{7}{3} \][/tex]
Now, divide both sides by 7:
[tex]\[ x = -\frac{7}{3} \div 7 \][/tex]
[tex]\[ x = -\frac{7}{3} \times \frac{1}{7} \][/tex]
[tex]\[ x = -\frac{1}{3} \][/tex]
### Step 4: Substitute [tex]\(x\)[/tex] back into the expression for [tex]\(y\)[/tex]
Now that we have [tex]\(x = -\frac{1}{3}\)[/tex], we substitute this value back into the expression for [tex]\(y\)[/tex]:
[tex]\[ y = x + \frac{5}{3} \][/tex]
[tex]\[ y = -\frac{1}{3} + \frac{5}{3} \][/tex]
Simplify the fraction:
[tex]\[ y = \frac{-1 + 5}{3} \][/tex]
[tex]\[ y = \frac{4}{3} \][/tex]
### Conclusion
The solution to the system of equations is:
[tex]\[ x = -\frac{1}{3}, \quad y = \frac{4}{3} \][/tex]
So the solution is:
[tex]\[ \left( x, y \right) = \left( -\frac{1}{3}, \frac{4}{3} \right) \][/tex]
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