Substitution 19793

Calculate in arithmetic sequence a1, d, s7,
if:
a1 + a4 + a6 = 71
a5 - a3 - a2 = 2

Hint: Use the substitution method when solving the system.
Pay due attention to the "minus" signs in the second equation of the system.

Correct answer:

a₁ =  5
d =  7
a₇ =  47

Step-by-step explanation:

$$\begin{gathered} a_1+a_4+a_6=71 \\ {a}_5-a_3-a_2=2 \\ {a}_1+(a_1+3d)+(a_1+5d)=71 \\ {a}_1+4d-(a_1+2d)-(a_1+d)=2 \\ {a}_1+(a_1+3\cdot d)+(a_1+5\cdot d)=71 \\ {a}_1+4\cdot d-(a_1+2\cdot d)-(a_1+d)=2 \\ 3a_1+8d=71 \\ {a}_1-d=-2 \\ Row2-\frac{1}{3}\cdot Row1\to Row2 \\ 3a_1+8d=71 \\ -3.67d=-25.67 \\ d=\frac{-25.66666667}{-3.66666667}=7 \\ {a}_1=\frac{71-8d}{3}=\frac{71-8\cdot 7}{3}=5 \\ {a}_1=5 \\ d=7 \end{gathered}$$

$d=7$

$a_7=a_1+6\cdot d=5+6\cdot 7=47$

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