Question 835353
USING JUST ONE VARIABLE:
{{{x}}}= the smaller number
{{{x+7}}}= the larger number (since the difference between them is 7)
{{{x+(x+7)=25}}} is our equation (because the sum of the two numbers is 25).
Solving:
{{{x+(x+7)=25}}}
{{{(x+x)+7=25}}}
{{{2x+7=25}}}
{{{2x=25-7}}}
{{{2x=18}}}
{{{x=18/2}}}
{{{highlight(x=9)}}} is the smaller number.
Then, {{{x+7=9+7=highlight(16)}}} is the larger number.
The two numbers are {{{highlight(9)}}} and {{{highlight(16)}}} .
 
USING TWO VARIABLES:
{{{x}}}= the smaller number
{{{y}}}= the larger number
The equations are
{{{x+y=25}}} (because the sum of the two numbers is 25), and
{{{y-x=7}}} (because the difference of the two numbers is 7).
We can solve the system {{{system(x+y=25,y-x=7)}}} by substitution by solving
{{{y-x=7}}} for {{{y}}} , getting {{{y=x+7}}} and then substituting {{{x+7}}} for {{{y}}} in {{{x+4=25}}} .
Then we get the one variable equation solved above.
We can also solve the system {{{system(x+y=25,y-x=7)}}} by graphing, or by the "method" that some call elimination. (Others call it combinations of equations.
For example, adding both equations, we get
{{{y+x=25}}}
{{{y-x=7}}}
------------
{{{y+y=25+7}}} --> {{{2y=32}}} --> {{{y=32/2}}} ---> h{{{highligt(y=16)}}}
Then we can substitute that value for {{{y}}} into either of the equations to find {{{x}}} , as in
{{{x+y=25}}} --> {{{x+16=25}}} --> {{{x=25-16}}} -->  {{{highlight(x=9)}}} .
To solve by graphing we graph the line that represents {{{x+y=25}}} and the line that represents {{{y-x=7}}} . The coordinates of the point where they intersect is the (x,y) solution . We estimate it (read it) from the graph, to get the values of x and y that appear to be the solution, and then verify by plugging those values into the equations.
Graphing is a bit time consuming.
To graph one linear equation, we just need to points (or a pont and a slope).
For {{{x+y=25}}} , we can see that (0,25) and {25,0) are two points in the graph, because when we set x to zero, the equation turns into {{{y=25}}} , and hen we set y to zero, the equation turns into {{{x=25}}} . The points (0,25) and (25,0) are the y-intercept and x-intercept, and we can use them to draw the line.
For {{{y-x=7}}} we can find the points (0,7) and (-7,0) in a similar way.
{{{drawing(500,500,-10,30,-10,30,
grid(1),
red(circle(0,25,0.3)),red(circle(25,0,0.3)),
red(line(-10,35,30,-5)),locate(18.2,8,red(x+y=25)),
green(line(-10,-3,30,37)),locate(21,28,green(y-x=7)),
green(circle(0,7,0.3)),green(circle(-7,0,0.3))
)}}} The intersection point coordinates appear to be {{{x=9}}} and {{{y=16}}} .
We verify by substituting into {{{x+y=25}}} to get
{{{x+y=9+16=25}}} .
We also need to verify that Those values make {{{y=x=7}}} true, so we calculate
{{{y-x=16-9=7}}}.
With the values read from the graph verified, we can say that the solution found by graphing is
{{{highlight(system(x=9,y=16))}}} .