Question 82049
The solution to the system means that the point (-3, 4) satisfies both of the equations.
So you can substitute these values for x and y (x equals -3 and y equals +4) into both of 
the equations, and solve for the values of b and m.  
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First go to the equation:
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5x + 7y = b
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Substitute -3 for x and + 4 for y.  When you do that the equation becomes:
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5(-3) + 7(4) = b
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Multiply out the left side to get:
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-15 + 28 = b
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Add the two terms on the left side and the equation becomes 13 = b.  You now know the 
value of b.
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Next go to the second equation:
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mx + y = 22
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Again substitute -3 for x and +4 for y.  When you do the equation becomes:
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m(-3) + 4 = 22
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Get rid of the +4 on the left side by subtracting 4 from both sides.  When you do that
the equation simplifies to:
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m(-3) = 18
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Now solve for m by dividing both sides of the equation by -3 (the multiplier of m). When 
you do that division the equation becomes:
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m = 18/-3 = -6
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So the answers to the problem are b = 13 and m = -6
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A way you can check these two values is to put them into the original equations and solve 
for the set to see if the common solution is (-3, 4).  In other words, solve the set:
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5x + 7y = 13 and
-6x + y = 22
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Let's do it by variable elimination.  Multiply the top equation (all terms on both sides)
by 6 and the bottom equation (all terms on both sides) by 5. When you do that the 
equation set becomes:
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+30x + 42y =  78
-30x +  5y = 110
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Add the columns in these two equations. Note that this causes the +30x and the -30x to
cancel. The remaining additions in columns results in:
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47y = 188
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Solve for y by dividing both sides of this equation by 47 to get:
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y = 188/47 = 4
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Now return to either of the equations in which the values of b and m had been substituted.
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Let's select the equation -6x + y = 22.  We know that y = 4, so substitute that value for y
and solve for x.  Substituting for y the equation becomes:
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-6x + 4 = 22
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Subtract 4 from both sides:
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-6x = 18
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and divide both sides by -6 to get: x = 18/-6 = -3.  So the coordinate point that satisfies
the equation set is (-3, + 4) just as the problem said it was.  So our answers for m and b
check out.
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Hope this gives you some insight into the problem and how to solve it
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