Find $a$ and $b$ such that \[\frac{3x+5}{x^2-x-42}=\frac{a}{x- 7}+\frac{b}{x+6}.\]write your answer in the form $(a, b)$.

Assume

Combining the fractions on the right hand side gives

The fractions will be equal as long as the their numerator are equal:

Polynomials are equal to one another if the coefficients of like terms are equal:

Solving this, you'd get and , so that your answer would be (26, -23).

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Instead of solving the system of equations above, there is a trick that involves picking so that some terms disappear and solving for either is much faster.

At the point where we have

notice that setting will eliminate . Doing so, we get

while setting would give

last one.

step-by-step explanation:

last one.

generally speaking, the others follow some sort of pattern that correlates to either linear (looks like a straight line) or sinusoidal (looks like an s).

the last one is just a random cloud of points that doesn't correlate to any particular known order.

x = 2

step-by-step explanation:

if sr parallel to ba then the sides bc and ac are in proportion, that is

= ← substitute values

= ( cross- multiply )

27x = 6(4x + 1)

27x = 24x + 6 ( subtract 24x from both sides )

3x = 6 ( divide both sides by 3 )

x = 2