How to draw two linear lines and find where they cross without using the graph.

So, for a standard liner function, Y=mx+c, where Y is the output, m is the gradient or slope of our line, x is the input value and c is our y-intercept, where our line crosses the Y-axis. So, we have two equations, equation 1 which is Y=2x+2 and equation 2 which is Y=-x+5. We can look at these and find m and c by looking at them.Equation 1 has gradient 2 and y-intercept 2. Equation 2 has gradient -1 and y-intercept 5.To next draw our lines, we want the x-intercept, so we make y equal 0. This is because the x-axis and y axis cross each other when they equal 0.So, in equation 1, we have 0=2x+2. We can take away two from both sides, so we have -2=2x. Then we divide by two, as we want to find what x is on its own. so we have -1=x, or x=-1.In equation 2, with Y=0, we have 0=-x+5. This is simpler, as we add x to both sides, so we get x=5 as our x-intercept.
We now need to draw our axis, so as a standard, I draw from -5 to 5 for both axis, crossing at (0,0).To draw our two lines, we mark points (-1,0) and (2,0) for equation 1 and then connect the lines extending them out both ways to the maximums of our axis.We then do the same for equation two, marking (5,0) and (0,5) for our intercepts and again extend them to fill the axis we have drawn.From this, we can approximate the crossing of the two lines. Find where they are crossed and then use a ruler to draw dotted lines to the axis . we should find that they cross at (1,4).
To find this answer, we can also do simultaneous equations.To begin this we need our two equations again, Y=2x+2 and Y=-x+5.Doing simultaneous equations, we want to eliminate a variable, so the Y or the x, so we have x=... or Y=..., by making them equal to each other.Here, it is easier to cancel the Y's, so we substitute into our equation 1, -x+5 for the Y.So, we have -x+5=2x+2.we want all of the x's on one side, so we add x to both sides, giving us 5=3x+2.we then want all of the numbers on one side, so take away 2 from both sides.we get 3=3x. We can divide by three, as we want x=..., to give x=1.We can then put this back into an equation instead of an x to give us our Y=....so, into equation , we now have Y=2(1)+2. We use the brackets to show where the x was and that this needs to be multiplied by 2.So we get Y=2+2, so Y=4. The Lines cross at (1,4) which is what we got from our drawing.