SOLUTION: Find the values for m and b in the following system so that the solution to the system is (-3,4) 5x+7y=b mx+y=22 Please, and thank you.

Question 82049This question is from textbook beginners algebra
: Find the values for m and b in the following system so that the solution to the system is (-3,4)
5x+7y=b
mx+y=22
Please, and thank you. This question is from textbook beginners algebra

Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
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.
.
First go to the equation:
.
5x + 7y = b
.
Substitute -3 for x and + 4 for y. When you do that the equation becomes:
.
5(-3) + 7(4) = b
.
Multiply out the left side to get:
.
-15 + 28 = b
.
Add the two terms on the left side and the equation becomes 13 = b. You now know the
value of b.
.
Next go to the second equation:
.
mx + y = 22
.
Again substitute -3 for x and +4 for y. When you do the equation becomes:
.
m(-3) + 4 = 22
.
Get rid of the +4 on the left side by subtracting 4 from both sides. When you do that
the equation simplifies to:
.
m(-3) = 18
.
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:
.
m = 18/-3 = -6
.
So the answers to the problem are b = 13 and m = -6
.
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:
.
5x + 7y = 13 and
-6x + y = 22
.
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:
.
+30x + 42y = 78
-30x + 5y = 110
.
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:
.
47y = 188
.
Solve for y by dividing both sides of this equation by 47 to get:
.
y = 188/47 = 4
.
Now return to either of the equations in which the values of b and m had been substituted.
.
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:
.
-6x + 4 = 22
.
Subtract 4 from both sides:
.
-6x = 18
.
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.
.
Hope this gives you some insight into the problem and how to solve it
.
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