```Question 184719
- Determine how many solutions exist
- Use either elimination or substitution to find the solutions (if any)
Graph the two lines, labeling the x-intercepts, y-intercepts, and points of intersection
0.2x + 0.4y = 1.7 and 8.3x - 6.3y = -4.3
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get rid of the decimals, multiply both equations by 10, results
2x + 4y = 17
83x - 63y = -43
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We will use elimination;
Multiply the 1st equation by 83 and the 2nd equation by 2:
166x + 332y = 1411
166x - 126y = -86
--------------------subtraction eliminates x, find y
458y = 1497
y = {{{1497/458}}}; approx 3.3 on the graph
Find x using eq: 2x + 4y = 17
2x + 4({{{1497/458}}}) = 17
2x + {{{5988/458}}} = 17
2x = -{{{5988/458}}} + 17
2x = -{{{5988/458}}} + {{{7786/458}}}
2x = {{{1798/458}}}
x = {{{1/2}}}*{{{1798/458}}}
x = {{{1798/916}}}; approx 2.0 on the graph
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solution: x = {{{1798/458}}}; y = {{{1497/458}}}; this the point of intersection
:
Put both equations in the slope intercept form
2x + 4y = 17
4y = -2x + 17
y = {{{-2/4}}}x + {{{17/4}}}
y = {{{-1/2}}}x + {{{17/4}}}
Plot two points
x = 4; y = 2{{{1/4}}} (substituted 4 for x in the above equation and found y)
x = 0; y = {{{17/4}}}; (4.25) this is also the y intercept (when x=0)
Graph of this
{{{ graph( 300, 200, -6, 12, -8, 10, -.5x+(17/4)) }}}
calculate the x intercept (when y = 0)
{{{-1/2}}}x + {{{17/4}}} = 0
{{{-1/2}}}x = - {{{17/4}}}
Multiply both side by -2, results
x = {{{34/4}}} = 8{{{1/2}}} is the x intercept, confirmed on the graph
:
Do the same with the 2nd equation;
83x - 63y = -43
-63y = -83x - 43
63y = 83x + 43; multiplied by -1
y = {{{83/63}}}x + {{{43/63}}}
Plot two points
x = 0, y = {{{43/63}}}; approx .7 on the graph; this the y intercept
x = 4, y = {{{375/63}}}; approx 6 on the graph
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Add this to the above graph:
{{{ graph( 300, 200, -6, 12, -8, 10, -.5x+(17/4), (83/63)x+(43/63))) }}}
:
Calculate the x intercept of this equation:
{{{83/63}}}x + {{{43/63}}} = 0
{{{83/63}}}x = -{{{43/63}}}
Multiply both sides by 63
83x = 43
x = {{{-43/83}}}; approx -.5 on the graph, (the x intercept

Summarize
Eq1: x intercept = 8.5, y intercept = 4.25
Eq2: x intercept = {{{-43/83}}}, y intercept = {{{43/63}}}
Point of intersection: x = {{{1798/458}}}; y = {{{1497/458}}} (approx 2, 3.3 on the graph
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This problems was very time consuming, please take time to study it and learn
what's happening here.  Carl```