(x+4)/3-x/7 =(x+7)/5 — step-by-step solution

1) Let's solve in

ℝ

\frac{x + 4}{3} - \frac{x}{7} = \frac{x + 7}{5}

2) Let's add fractions:

\frac{x + 4}{3} - \frac{x}{7} - \frac{x + 7}{5} = 0

ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{x + 4}{3} - \frac{x}{7}

\frac{(x + 4) \cdot 7 - x \cdot 3}{3 \cdot 7}

\frac{(x + 4) \cdot 7 - x \cdot 3}{3 \cdot 7} - \frac{x + 7}{5} = 0

ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{(x + 4) \cdot 7 - x \cdot 3}{21} - \frac{x + 7}{5}

\frac{((x + 4) \cdot 7 - x \cdot 3) \cdot 5 - (x + 7) \cdot 21}{21 \cdot 5}

ⅰ

3 \cdot 7

21

\frac{((x + 4) \cdot 7 - x \cdot 3) \cdot 5 - (x + 7) \cdot 21}{21 \cdot 5} = 0

ⅰ

21 \cdot 5

(20 + 1) \cdot 5

20 \cdot 5 + 5

100 + 5

105

\frac{((x + 4) \cdot 7 - x \cdot 3) \cdot 5 - (x + 7) \cdot 21}{105} = 0

Let's solve in

ℝ

((x + 4) \cdot 7 - x \cdot 3) \cdot 5 - (x + 7) \cdot 21 = 0

ⓘ Expand by multiplying each term inside the parentheses:

(x + 4) \cdot 7

x \cdot 7 + 28

((x \cdot 7 + 28) - x \cdot 3) \cdot 5

(x \cdot 7 + 28) \cdot 5 + 15 \cdot (- x)

- (x + 7) \cdot 21

- (x \cdot 21 + 147)

(x \cdot 7 + 28) \cdot 5 + 15 \cdot (- x) - (x \cdot 21 + 147) = 0

ⓘ The minus sign is placed in front of the product:

15 \cdot (- x)

- 15 x

■A minus sign before a parenthesis changes the sign of each of its terms:

- (x \cdot 21 + 147)

- x \cdot 21 - 147

(x \cdot 7 + 28) \cdot 5

- 15 x

- x \cdot 21

- 147 = 0

ⓘ Like terms are grouped by adding their coefficients:

(x \cdot 7 + 28) \cdot 5 - 15 x - x \cdot 21

5 \cdot (x \cdot 7 + 28) - 36 x

5 \cdot (x \cdot 7 + 28) - 36 x - 147 = 0

ⓘ Expand by multiplying each term inside the parentheses:

5 \cdot (x \cdot 7 + 28)

35 x + 140

35 x

+ 140

- 36 x

- 147 = 0

ⓘ Start from

140

and move back

147

on the number line, going below zero:

140 - 147

-7

■Like terms are grouped by adding their coefficients:

35 x - 36 x

- x

- x - 7 = 0

ⓘ We add

7

to both sides

- x = 7

ⓘ The sign of both sides is changed

▶Solution found:

x = -7