2x+y=2, 3x-y=4

1) We have the 2 equations:

[1]:

2 x + y = 2

[2]:

3 x - y = 4

2) Let's solve for x:

2 x + y = 2

[1]

2 x = 2 - y

ⓘ Divide both sides by

2

Solution found [1]:

x = \frac{2 - y}{2}

3) Let's solve for y:

3 x - y = 4

[2] using

x = \frac{2 - y}{2}

ⓘ Multiplying a fraction by a number multiplies its numerator:

3 \cdot \frac{2 - y}{2}

\frac{3 \cdot (2 - y)}{2}

\frac{3 \cdot (2 - y)}{2} - y = 4

4) Let's sum the fractions:

\frac{3 \cdot (2 - y)}{2} - y - 4 = 0

ⓘ Add fractions by using a common denominator and then adding their numerators:

\frac{3 \cdot (2 - y)}{2} - y

\frac{3 \cdot (2 - y) - y \cdot 2}{2}

\frac{3 \cdot (2 - y) - y \cdot 2}{2} - 4 = 0

ⓘ Add fractions by using a common denominator and then adding their numerators:

\frac{3 \cdot (2 - y) - y \cdot 2}{2} - 4

\frac{(3 \cdot (2 - y) - y \cdot 2) - 4 \cdot 2}{2}

\frac{(3 \cdot (2 - y) - y \cdot 2) - 4 \cdot 2}{2} = 0

ⓘ Perform multiplications before additions and subtractions:

4 \cdot 2

8

\frac{3 \cdot (2 - y) - y \cdot 2 - 8}{2} = 0

⇥

3 \cdot (2 - y) - y \cdot 2 - 8 = 0

⇥ⓘ Expand using the distributive law, multiplying each term of the sum by the other factor:

3 \cdot (2 - y)

6 + 3 \cdot (- y)

⇥

6

+ 3 \cdot (- y)

- y \cdot 2

- 8 = 0

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

3 \cdot (- y)

- 3 y

■Start from

6

and move back

8

on the number line, going below zero:

6 - 8

-2

⇥

- 2 - 3 y - y \cdot 2 = 0

⇥ⓘ Like terms are grouped by adding their coefficients:

- 3 y - y \cdot 2

-5 y

⇥

- 2 - 5 y = 0

⇥ⓘ Add

2

to both sides

⇥

- 5 y = 2

⇥ⓘ Divide both sides by

-5

⇥

y = \frac{2}{-5}

⇥ⓘ The sign of a fraction goes in front of the bar, whether it comes from the numerator or the denominator:

\frac{2}{-5}

- \frac{2}{5}

⇥Solution found [2]:

y = - \frac{2}{5}

5) We find the value of x using:

x = \frac{2 - y}{2}

using

y = - \frac{2}{5}

x = \frac{2 + \frac{2}{5}}{2}

ⓘ Add fractions by using a common denominator and then adding their numerators:

2 + \frac{2}{5}

\frac{5 \cdot 2 + 2}{5}

x = \frac{\frac{5 \cdot 2 + 2}{5}}{2}

ⓘ Perform multiplications before additions and subtractions:

5 \cdot 2

10

x = \frac{\frac{10 + 2}{5}}{2}

ⓘ Calculate the numerator and denominator before dividing:

10 + 2

12

x = \frac{\frac{12}{5}}{2}

ⓘ Dividing a fraction by

2

is multiplying its denominator by

2

\frac{\frac{12}{5}}{2}

\frac{12}{5 \cdot 2}

■To simplify a fraction, divide its numerator and denominator by the same nonzero number:

\frac{12}{5 \cdot 2}

\frac{6}{5}

Solution found:

x = \frac{6}{5}

▶Solution:

{x = \frac{6}{5}, y = - \frac{2}{5}}

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