# One number is 4 less than 3 times a second number. If 3 more than two times the first number is decreased by 2 times the second number, the result is 11. Use the substitution method. What is the first number?

Jun 16, 2016

${n}_{1} = 8$
${n}_{2} = 4$

#### Explanation:

One number is 4 less than -> n_1=?-4
3 times " ........................."->n_1=3?-4
the second number $\textcolor{b r o w n}{\text{..........} \to {n}_{1} = 3 {n}_{2} - 4}$
$\textcolor{w h i t e}{\frac{2}{2}}$

If 3 more " ..........................................."-> ?+3
than two times the first number$\text{ ............} \to 2 {n}_{1} + 3$
is decreased by "..................................."->2n_1+3-?
2 times the second number$\text{ .................} \to 2 {n}_{1} + 3 - 2 {n}_{2}$
the result is 11$\textcolor{b r o w n}{\text{ .....................................} \to 2 {n}_{1} + 3 - 2 {n}_{2} = 11}$

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$\textcolor{w h i t e}{.} {n}_{1} = 3 {n}_{2} - 4$ .....................Equation (1)
$2 {n}_{1} + 3 - 2 {n}_{2} = 11$...........Equation (2)

Substitute for ${n}_{1}$ in equation (2) using equation (1)

$\textcolor{b r o w n}{2 {n}_{1} + 3 - 2 {n}_{2} = 11} \text{ } \textcolor{b l u e}{\to 2 \left(3 {n}_{2} - 4\right) + 3 - 2 {n}_{2} = 11}$

Multiplying out the brackets

$6 {n}_{2} - 8 + 3 - 2 {n}_{2} = 11$

$4 {n}_{2} - 5 = 11$

Add 5 to both sides

$4 {n}_{2} = 16$

Divide both sides by 4

$\text{ } \textcolor{g r e e n}{{n}_{2} = 4}$
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Substitute ${n}_{2} = 4$ into equation (1)

${n}_{1} = 3 \left(4\right) - 4$

${n}_{1} = 12 - 4$

$\text{ } \textcolor{g r e e n}{{n}_{1} = 8}$
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Check: One number is 4 less than 3 times the second number

$8 = 3 \left(4\right) - 4 = 8$ confirmed!