Answer :

GinnyAnswer
To solve the given system of linear equations:

[tex]\[ \begin{cases} x + 2y = 1 \\ -3x + y = 10 \end{cases} \][/tex]

we will use the method of substitution or elimination. Here is a step-by-step solution using the elimination method:

1. Label the equations for reference:
[tex]\[ \text{(1) } x + 2y = 1 \][/tex]
[tex]\[ \text{(2) } -3x + y = 10 \][/tex]

2. Eliminate one variable by manipulating the equations. We can eliminate [tex]\( y \)[/tex] by making the coefficients of [tex]\( y \)[/tex] in both equations match.

Let’s multiply the second equation by 2 to match the coefficient of [tex]\( y \)[/tex] in the first equation:

[tex]\[ \text{(2) } -3x + y = 10 \quad \Rightarrow \quad -6x + 2y = 20 \][/tex]

Now, we have:
[tex]\[ \begin{cases} x + 2y = 1 \quad \text{(Equation 1)} \\ -6x + 2y = 20 \quad \text{(Modified Equation 2)} \end{cases} \][/tex]

3. Subtract Equation 1 from the Modified Equation 2 to eliminate [tex]\( y \)[/tex]:
[tex]\[ (-6x + 2y) - (x + 2y) = 20 - 1 \][/tex]

Simplifying:
[tex]\[ -6x + 2y - x - 2y = 19 \][/tex]
[tex]\[ -7x = 19 \][/tex]

4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{19}{-7} \][/tex]
[tex]\[ x = -\frac{19}{7} \][/tex]

5. Substitute the value of [tex]\( x \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]. We will use Equation 1:
[tex]\[ x + 2y = 1 \][/tex]
[tex]\[ -\frac{19}{7} + 2y = 1 \][/tex]

6. Isolate [tex]\( y \)[/tex]:
[tex]\[ 2y = 1 + \frac{19}{7} \][/tex]
[tex]\[ 2y = \frac{7}{7} + \frac{19}{7} \][/tex]
[tex]\[ 2y = \frac{26}{7} \][/tex]
[tex]\[ y = \frac{26}{7 \cdot 2} \][/tex]
[tex]\[ y = \frac{26}{14} \][/tex]
[tex]\[ y = \frac{13}{7} \][/tex]

So, the solution to the system of equations is:
[tex]\[ \boxed{\left( -\frac{19}{7}, \frac{13}{7} \right)} \][/tex]

Other Questions