# How do I use matrices to find the solution of the system of equations 3x+4y=10 and x-y=1?

Aug 19, 2014

3 ways, Cramer's Rule, Elimination, or Substitution. Let's look at Cramer's rule below:

Standard equation $1 = a x + b y = c$ and Standard equation $2 = \mathrm{dx} + e y = f$

Therefore:
$a = 3 , b = 4 , c = 10 , d = 1 , e = - 1 , f = 1$

Step 1, calculate the denominator Delta ($\Delta$):

$\Delta = a \cdot e - b \cdot d$
$\Delta = \left(3 \cdot - 1\right) - \left(4 \cdot 1\right)$
$\Delta = - 3 - 4$
$\Delta = - 7$

Step 2, calculate the numerator for $x$:

${N}_{x} = c \cdot e - b \cdot f$
${N}_{x} = \left(10 \cdot - 1\right) - \left(4 \cdot 1\right)$
${N}_{x} = - 10 - 4$
${N}_{x} = - 14$

Step 3, calculate the numerator for $y$:

${N}_{y} = a \cdot f - c \cdot d$
${N}_{y} = \left(3 \cdot 1\right) - \left(10 \cdot 1\right)$
${N}_{y} = 3 - 10$
${N}_{y} = - 7$

Now we have all of our components. Evaluate and solve:

$x = {N}_{x} / \Delta$

$x = - \frac{14}{-} 7$

$x = 2$

$y = {N}_{y} / \Delta$

$y = - \frac{7}{-} 7$

$y = 1$

For calculator help with similar problems, check out the 2 unknowns calculator

Just wanted to mention that $\Delta , {N}_{x} , {N}_{y}$ are called determinants, in case anyone wants to look up more info about matrices.