# How do you solve the system of equations 3x + y = 23 and 4x - y = 19?

Jan 7, 2018

X=6, Y=5

#### Explanation:

Use simultaneous equations (multiplying both equations, and eliminating either x or y by adding or subtracting to find the value of one, then plugging the value back in to find the other value)

The lowest common multiple of 3 and 4 is 12
$3 \cdot 4 = 12$ so we have to multiply the first equation by 4
$4 \cdot 3 = 12$ so we have to multiply the second equation by 3

This is the result:
$12 x + 4 y = 92$
$12 x - 3 y = 57$

From here as x is the same, you can subtract the bottom equation from the top leaving you with y. After doing this the result is:

$7 y = 35$
$y = \frac{35}{7} = 5$

Then we can plug this value back into the equations, then we can solve to find x.
$3 x + y = 23$
$3 x + 5 = 23$
$3 x = 18$ (-5)
$x = \frac{18}{3} = 6$

We have got $x = 6$ and $y = 5$, but we have to check if these values work for both equations so lets check.

$4 x - y = 19$
$4 \cdot 6 - 5 = 19$
$24 - 5 = 19$

So therefore these values fit both equations and we are correct.

Jan 7, 2018

The point of intersection is $\left(6 , 5\right)$

#### Explanation:

You can also use substitution. The resulting values for $x$ and $y$ are the point of intersection of the two equations.

Equation 1: $3 x + y = 23$

Equation 2: $4 x - y = 19$

Solve Equation 1 for $y$.

$3 x + y = 23$

Subtract $3 x$ from both sides.

$y = 23 - 3 x$

Substitute $23 - 3 x$ for $y$ in Equation 2 and solve for $x$.

$4 x - y = 19$

$4 x - \left(23 - 3 x\right) = 19$

Simplify.

$4 x - 23 + 3 x = 19$

Add $23$ to both sides.

$4 x + 3 x = 19 + 23$

Simplify.

$7 x = 42$

Divide both sides by $7$.

$x = \frac{42}{7}$

$x = 6$

Substitute $6$ for $x$ into Equation 1 and solve for $y$.

$3 x + y = 23$

$3 \left(6\right) + y = 23$

Simplify.

$18 + y = 23$

Subtract $18$ from both sides.

$y = 23 - 18$

$y = 5$

The point of intersection is $\left(6 , 5\right)$.

Equation 1:

$3 \left(6\right) + 5 = 23$

$18 + 5 = 23$

$23 = 23$ sqrt

Equation 2:

$4 \left(6\right) - 5 = 19$

$24 - 5 = 19$

$19 = 19$ sqrt

graph{(y+3x-23)(-y+4x-19)=0 [-15.69, 16.34, -5.83, 10.19]}