Question 664547
A triangle is formed by the intersection of the lines

{{{2x + 3y = 14}}}, 

{{{4x - 5y = -16}}}, 

and the {{{x - axis}}}

Find the area of the triangle. 
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first graph the lines, using two points ({{{x}}} and {{{y}}} intercepts)

{{{4x - 5y = -16}}},

set {{{x=0}}}, find {{{y-intercept}}}

{{{4*0 - 5y = -16}}}.........{{{16 = 5y}}}..=>..{{{16/5 = y}}}.=>..{{{3.2 = y}}}

one point is: ({{{0}}},{{{3.2}}}


set {{{y=0}}}, find {{{x-intercept}}}

{{{4x - 5*0 = -16}}}.........{{{4x = -16}}}..=>..{{{x=-16/4}}}.=>..{{{x=-4}}}

one point is: ({{{-4}}},{{{0}}}


same with other line: {{{2x + 3y = 14}}}

set {{{x=0}}}, find {{{y-intercept}}}

{{{2*0+ 3y = 14}}}.........{{{3y= 14}}}..=>..{{{y= 14/3}}}.=>..{{{y=4.67}}}

one point is: ({{{0}}},{{{4.67}}})


set {{{y=0}}}, find {{{x-intercept}}}

{{{2x+ 3*0 = 14}}}.........{{{2x =14}}}..=>..{{{x=7}}}

one point is: ({{{7}}},{{{0}}})


graph them:

{{{drawing(600, 600, -10, 10, -10, 10,grid(1),circle(-4,0,0.2),circle(7,0,0.2),circle(1,4,0.2),graph(600, 600, -10, 10, -10, 10,-(2/3)x+14/3, (4/5)x+16/5))}}}


as you can see from the graph, the length of the base is {{{b=11}}} units, and the length of the height {{{h=4}}} units (lines intersect in a point ({{{1}}},{{{4}}}) ; so, the area of the triangle will be:


{{{A=(1/2) b*h}}} 

{{{A=(1/2) 11units*4units}}}

{{{A=(1/cross(2)1) 11units*cross(4)2units}}}

{{{A=11*2units^2}}}

{{{A=22units^2}}}