Answer :
To solve the system of equations:
[tex]\[ \left\{\begin{array}{l} -2 x+3 y=13 \\ 3 x+4 y=6 \end{array}\right. \][/tex]
We'll use the method of elimination or substitution to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
1. Step 1: Write down the system of equations:
[tex]\[ \left\{\begin{array}{l} -2x + 3y = 13 \\ 3x + 4y = 6 \end{array}\right. \][/tex]
2. Step 2: Choose one of the equations and solve for one variable. Let's solve the first equation for [tex]\(x\)[/tex]:
[tex]\[ -2x + 3y = 13 \][/tex]
We can solve for [tex]\(x\)[/tex] as follows:
[tex]\[ -2x = 13 - 3y \\ x = \frac{13 - 3y}{-2} \\ x = -\frac{13}{2} + \frac{3}{2}y \][/tex]
3. Step 3: Substitute this expression for [tex]\(x\)[/tex] into the second equation:
[tex]\[ 3\left(-\frac{13}{2} + \frac{3}{2}y\right) + 4y = 6 \][/tex]
Multiply through by 2 to clear the fraction:
[tex]\[ 3\left(-13 + 3y\right) + 8y = 12 \\ -39 + 9y + 8y = 12 \\ -39 + 17y = 12 \][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[ 17y = 12 + 39 \\ 17y = 51 \\ y = \frac{51}{17} \\ y = 3 \][/tex]
4. Step 4: Substitute the value of [tex]\(y\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex]. Use the first equation:
[tex]\[ -2x + 3(3) = 13 \\ -2x + 9 = 13 \\ -2x = 13 - 9 \\ -2x = 4 \\ x = \frac{4}{-2} \\ x = -2 \][/tex]
So, the solution to the system of equations is:
[tex]\[ (x, y) = (-2, 3) \][/tex]
Therefore, the correct choice among the provided options is:
[tex]$(-2, 3)$[/tex].
[tex]\[ \left\{\begin{array}{l} -2 x+3 y=13 \\ 3 x+4 y=6 \end{array}\right. \][/tex]
We'll use the method of elimination or substitution to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
1. Step 1: Write down the system of equations:
[tex]\[ \left\{\begin{array}{l} -2x + 3y = 13 \\ 3x + 4y = 6 \end{array}\right. \][/tex]
2. Step 2: Choose one of the equations and solve for one variable. Let's solve the first equation for [tex]\(x\)[/tex]:
[tex]\[ -2x + 3y = 13 \][/tex]
We can solve for [tex]\(x\)[/tex] as follows:
[tex]\[ -2x = 13 - 3y \\ x = \frac{13 - 3y}{-2} \\ x = -\frac{13}{2} + \frac{3}{2}y \][/tex]
3. Step 3: Substitute this expression for [tex]\(x\)[/tex] into the second equation:
[tex]\[ 3\left(-\frac{13}{2} + \frac{3}{2}y\right) + 4y = 6 \][/tex]
Multiply through by 2 to clear the fraction:
[tex]\[ 3\left(-13 + 3y\right) + 8y = 12 \\ -39 + 9y + 8y = 12 \\ -39 + 17y = 12 \][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[ 17y = 12 + 39 \\ 17y = 51 \\ y = \frac{51}{17} \\ y = 3 \][/tex]
4. Step 4: Substitute the value of [tex]\(y\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex]. Use the first equation:
[tex]\[ -2x + 3(3) = 13 \\ -2x + 9 = 13 \\ -2x = 13 - 9 \\ -2x = 4 \\ x = \frac{4}{-2} \\ x = -2 \][/tex]
So, the solution to the system of equations is:
[tex]\[ (x, y) = (-2, 3) \][/tex]
Therefore, the correct choice among the provided options is:
[tex]$(-2, 3)$[/tex].
