Simultaneous Equations

Ap - Or = 5, Ap + 2Or = 2. Subtracting, -3Or = 3, so Or = -1. Then Ap = 4, and the sum = 3.

We let $x$ be the apple and $y$ be the orange. We have

$x-y=5$ $\color{#D61F06}(1)$

$x+2y=2$ $\color{#D61F06}(2)$

Method 1: By substitution

In $\color{#D61F06}(1)$ , we solve $x$ in terms of $y$ then substitute in $\color{#D61F06}(2)$ . We have

$x-y=5$ $\color{#D61F06}(1)$ $\implies$ $x=5+y$

Then substitute in $\color{#D61F06}(2)$ ,

$5+y+2y=2$

$5+3y=2$

$3y=-3$

$y=-1$

Now solve for $x$ using any equation,

$x=5+y$ $\implies$ $x=5-1=4$

$\text{Alternate Solution}$

In $\color{#D61F06}(1)$ , we solve $y$ in terms of $x$ then substitute in $\color{#D61F06}(2)$ . We have

$x-y=5$ $\implies$ $y=x-5$

Then substitute in $\color{#D61F06}(2)$

$x+2(x-5)=2$

$x+2x-10=2$

$3x=12$

$x=4$

Now solve for $y$ using any equation,

$y=x-5 = 4-5=-1$

Method 2: Elimination by addition or subtraction

Subtract $\color{#D61F06}(1)$ from $\color{#D61F06}(2)$ , we obtain

$3y=-3$ $\implies$ $y=-1$

Now solve for $x$ using any equation,

$x-(-1)=5$

$x+1=5$

$x=4$

Check by substituting the values to the original equations

$x-y=5$

$4-(-1)=5$

$4+1=5$

$5=5$ $okay$

$x+2y=2$

$4+2(-1)=2$

$4-2=2$

$2=2$ $okay$

Finally,

$x+y=4-1=$ $\boxed{3}$