Linear Equations and Systems – Algebra (No Calculator)

Solution

To isolate x, we first subtract 7 from both sides of the equation:
3x + 7 = 19
3x + 7 − 7 = 19 − 7
3x = 12

Next, we divide both sides by 3:
x = 12 ÷ 3 = 4

Therefore, x = 4. This problem is a straightforward example of solving a basic linear equation where you aim to isolate the variable by applying inverse operations step by step.

Solution

Distribute the 2 on the left side of the equation:
2(x − 4) = 6
2x − 8 = 6

Now, add 8 to both sides:
2x − 8 + 8 = 6 + 8
2x = 14

Finally, divide both sides by 2:
x = 14 ÷ 2 = 7

Thus, x = 7. Note how distributing a factor across terms inside parentheses is a common first step in many linear equations.

Solution

Start by isolating the term with x. Add 3 to both sides:
6x − 3 + 3 = 21 + 3
6x = 24

Now divide by 6:
x = 24 ÷ 6 = 4

The answer is x = 4. Checking the work: 6(4) − 3 = 24 − 3 = 21, which confirms the solution is correct.

Solution

Move the constant term to the other side:
-5x + 10 = 0
-5x = -10

Divide both sides by -5:
x = -10 / -5 = 2

Therefore, x = 2. Note that negative divided by negative is a positive value.

Solution

First, isolate the term with x by adding 4 to both sides:
(1/2)x − 4 + 4 = 5 + 4
(1/2)x = 9

Multiply both sides by 2 (the reciprocal of 1/2) to solve for x:
x = 9 × 2 = 18

The final answer is x = 18.

Solution

To solve for x, first move all x-terms to one side. Subtract 2x from both sides:
4x + 3 − 2x = 2x + 17 − 2x
2x + 3 = 17

Next, subtract 3 from both sides:
2x = 17 − 3
2x = 14

Finally, divide by 2:
x = 14 ÷ 2 = 7

Hence, x = 7.

Solution

First, distribute the 2:
2(x + 1) - x = 9
2x + 2 - x = 9

Combine like terms (2x - x):
x + 2 = 9

Subtract 2 from both sides:
x = 7

The result is x = 7.

Solution

Move y to the other side by subtracting 5 or rearrange to isolate y. Let’s isolate y:
5 - y = 12
-y = 12 - 5
-y = 7

Now multiply both sides by -1:
y = -7

Thus, y = -7. Always be careful with negative signs when isolating variables.

Solution

Subtract 5x from both sides to collect the x terms on one side:
7x - 5x - 2 = 5x - 5x + 10
2x - 2 = 10

Add 2 to both sides:
2x = 12

Divide by 2:
x = 6

Therefore, x = 6.

Solution

Subtract 3 from both sides to isolate the fraction term:
m/2 = 1 - 3
m/2 = -2

Multiply both sides by 2:
m = -2 × 2 = -4

Hence, m = -4.

Solution

Distribute 3:
3(2x - 5) = 9
6x - 15 = 9

Add 15 to both sides:
6x = 24

Divide by 6:
x = 4

Thus, x = 4.

Solution

First, bring the x-terms together by subtracting 3x from both sides:
4x - 3x + 2 = -5
x + 2 = -5

Subtract 2 from both sides:
x = -7

Therefore, x = -7.

Solution

Add 3x to both sides to get all the x-terms on one side:
8 = 5x + 13

Subtract 13 from both sides:
8 - 13 = 5x
-5 = 5x

Divide by 5:
x = -1

Hence, x = -1.

Solution

Expand the right side:
5 - 2x = 4 - 4x

Add 4x to both sides:
5 + 2x = 4

Subtract 5 from both sides:
2x = -1

Divide both sides by 2:
x = -1/2

Therefore, x = -1/2.

Solution

Multiply both sides by the reciprocal of (3/4), which is (4/3):
x = 6 × (4/3)

Simplify:
x = (6 × 4) / 3 = 24 / 3 = 8

Hence, x = 8.

Solution

First, simplify the term 9x/3:
9x/3 = 3x

So the equation becomes:
3x - 2 = 1

Add 2 to both sides:
3x = 3

Divide by 3:
x = 1

Therefore, x = 1.

Solution

Distribute -3:
10 - 3(2x + 1) = 1
10 - 6x - 3 = 1

Combine like terms (10 - 3):
7 - 6x = 1

Subtract 7 from both sides:
-6x = 1 - 7
-6x = -6

Divide by -6:
x = (-6)/(-6) = 1

Hence, x = 1.

Solution

Subtract 6 from both sides:
(2/3)x = 4

Multiply both sides by (3/2), the reciprocal of (2/3):
x = 4 × (3/2) = (4 × 3) / 2 = 12 / 2 = 6

Therefore, x = 6.

Solution

Expand the right side:
12 - x = 2x + 8

Add x to both sides to move x terms to the right:
12 = 3x + 8

Subtract 8 from both sides:
4 = 3x

Divide by 3:
x = 4/3

Therefore, x = 4/3 or 1.333... (but since no calculator is needed, 4/3 is a perfectly acceptable answer).

Solution

Distribute 5 on the left:
5(x - 3) = 2x + 10
5x - 15 = 2x + 10

Subtract 2x from both sides:
3x - 15 = 10

Add 15 to both sides:
3x = 25

Divide by 3:
x = 25/3

Hence, x = 25/3, which is approximately 8.333....

Solution

Distribute the minus sign across (x + 2):
7 - x - 2 = 4

Combine like terms:
5 - x = 4

Subtract 5 from both sides:
-x = -1

Multiply both sides by -1:
x = 1

Thus, x = 1.

Solution

Subtract x from both sides:
3x - x + 4 = 10
2x + 4 = 10

Subtract 4:
2x = 6

Divide by 2:
x = 3

Therefore, x = 3.

Solution

Subtract 9 from both sides:
-4x = 1 - 9 = -8

Divide both sides by -4:
x = -8 / -4 = 2

Hence, x = 2.

Solution

Subtract 5 from both sides:
x/4 = 2

Multiply both sides by 4:
x = 2 × 4 = 8

Therefore, x = 8.

Solution

Distribute 3:
-2x + 3x - 12 = -11

Combine like terms:
x - 12 = -11

Add 12 to both sides:
x = 1

Hence, x = 1.

Solution

One simple method is to add the two equations to eliminate y.
(x + y) + (x - y) = 5 + 1
2x = 6
x = 3

Substitute x = 3 into x + y = 5:
3 + y = 5
y = 2

The solution is (x, y) = (3, 2).

Solution

From the second equation, x - y = 1, we can isolate y:
y = x - 1

Substitute y = x - 1 into the first equation 2x + y = 8:
2x + (x - 1) = 8
3x - 1 = 8

Add 1 to both sides:
3x = 9

Divide by 3:
x = 3

Then y = 3 - 1 = 2

So, (x, y) = (3, 2).

Solution

From equation 2, x - 2y = -10, we can solve for x:
x = -10 + 2y

Substitute x = -10 + 2y into the first equation:
3(-10 + 2y) + 2y = 14
-30 + 6y + 2y = 14

Combine like terms:
-30 + 8y = 14

Add 30 to both sides:
8y = 44

Divide by 8:
y = 44/8 = 11/2 = 5.5

Now substitute y = 11/2 back into x = -10 + 2y:
x = -10 + 2(11/2) = -10 + 11 = 1

Hence, the solution is (x, y) = (1, 11/2).

Solution

Let’s use elimination. From the first equation, 2x - y = 4, we can solve for y:
-y = 4 - 2x
y = 2x - 4

Substitute y = 2x - 4 into the second equation:
3x + 2(2x - 4) = 1
3x + 4x - 8 = 1

Combine like terms:
7x - 8 = 1

Add 8 to both sides:
7x = 9

Divide by 7:
x = 9/7

Now substitute x = 9/7 into y = 2x - 4:
y = 2(9/7) - 4 = 18/7 - 4
y = 18/7 - 28/7 = -10/7

The solution is (x, y) = (9/7, -10/7).

Solution

Notice that adding these two equations will eliminate y:
(4x + y) + (4x - y) = 6 + 2
8x = 8
x = 1

Substitute x = 1 back into the first equation, 4(1) + y = 6:
4 + y = 6
y = 2

Thus, the solution is (x, y) = (1, 2).

Solution

Subtract equation 1 from equation 2 to eliminate y:
(3x + 2y) - (x + 2y) = 15 - 9
3x - x + 2y - 2y = 6
2x = 6
x = 3

Substitute x = 3 into x + 2y = 9:
3 + 2y = 9
2y = 6
y = 3

Therefore, (x, y) = (3, 3).

Solution

Observe that the second equation 10x - 6y = 4 is essentially twice the first equation 5x - 3y = 0, except for the constant term. Let’s see:
Multiply the first equation by 2:
(5x - 3y) * 2 = 0 * 2
10x - 6y = 0

Compare this to the second equation: 10x - 6y = 4. We have:
10x - 6y = 0 (from doubling eqn. 1)
10x - 6y = 4 (actual eqn. 2)

These two lines are parallel unless 4 = 0, which it does not. Hence, there is no solution. This is an inconsistent system.

Therefore, no solution (the system is inconsistent).

Solution

Since both right sides equal y, set them equal to each other:
2x + 1 = -x + 4

Add x to both sides:
3x + 1 = 4

Subtract 1:
3x = 3

Divide by 3:
x = 1

Substitute x = 1 into y = 2x + 1:
y = 2(1) + 1 = 3

The solution is (x, y) = (1, 3).

Solution

From x = 2y - 5, substitute into x + y = 7:
(2y - 5) + y = 7

Combine like terms:
3y - 5 = 7

Add 5 to both sides:
3y = 12

Divide by 3:
y = 4

Substitute y = 4 back into x = 2y - 5:
x = 2(4) - 5 = 8 - 5 = 3

Hence, (x, y) = (3, 4).

Solution

Notice that the second equation, 2x + y = 5, can be multiplied by 2 to match the first equation’s coefficients:
Multiply eqn. 2 by 2: 4x + 2y = 10

This is exactly the same as equation 1. Therefore, every solution that satisfies one also satisfies the other. The system represents the same line. Hence, there are infinitely many solutions:
Infinitely many solutions (the equations are dependent).

In slope-intercept form, from 2x + y = 5 => y = 5 - 2x. Any (x, y) pair that satisfies y = 5 - 2x is a solution.

Solution

Adding the equations to eliminate y:
(2x + 5y) + (3x - 5y) = 1 + 14
5x = 15
x = 3

Substitute x = 3 into 2x + 5y = 1:
2(3) + 5y = 1
6 + 5y = 1
5y = -5
y = -1

Thus, (x, y) = (3, -1).

Solution

From eqn. 1, x + 3y = 7, solve for x:
x = 7 - 3y

Substitute x = 7 - 3y into eqn. 2, 2x + y = 4:
2(7 - 3y) + y = 4

Distribute:
14 - 6y + y = 4

Combine like terms:
14 - 5y = 4

Subtract 14 from both sides:
-5y = -10

Divide by -5:
y = 2

Substitute y = 2 back into x = 7 - 3y:
x = 7 - 3(2) = 7 - 6 = 1

Hence, (x, y) = (1, 2).

Solution

Use substitution from eqn. 2, y = x - 1, into eqn. 1:
x - 4(x - 1) = 5

Distribute -4:
x - 4x + 4 = 5
-3x + 4 = 5

Subtract 4 from both sides:
-3x = 1

Divide by -3:
x = -1/3

Now find y:
y = x - 1 = -1/3 - 1 = -1/3 - 3/3 = -4/3

The solution is (x, y) = (-1/3, -4/3).

Solution

Adding the two equations eliminates y:
(3x + y) + (3x - y) = 11 + 5
6x = 16
x = 16/6 = 8/3

Substitute x = 8/3 into the first equation, 3x + y = 11:
3(8/3) + y = 11
8 + y = 11
y = 3

Hence, (x, y) = (8/3, 3).

Solution

From the first equation, x - 2y = -2, solve for x:
x = 2y - 2

Substitute into the second equation 2x + 3y = 11:
2(2y - 2) + 3y = 11

Distribute:
4y - 4 + 3y = 11

Combine like terms:
7y - 4 = 11

Add 4 to both sides:
7y = 15

Divide by 7:
y = 15/7

Then x = 2(15/7) - 2 = 30/7 - 14/7 = 16/7

So, (x, y) = (16/7, 15/7).

Solution

Substitute y = 2x into y + x = 9:
(2x) + x = 9
3x = 9
x = 3

Then y = 2(3) = 6

Hence, (x, y) = (3, 6).

Solution

Add the two equations to eliminate y:
(4x - y) + (4x + y) = 3 + 7
8x = 10
x = 10/8 = 5/4

Substitute x = 5/4 into 4x - y = 3:
4(5/4) - y = 3
5 - y = 3
-y = -2
y = 2

Therefore, (x, y) = (5/4, 2).

Solution

Substitute x = 2y - 2 into 3x + 4y = 16:
3(2y - 2) + 4y = 16
6y - 6 + 4y = 16
10y - 6 = 16

Add 6 to both sides:
10y = 22
y = 22/10 = 11/5

Now substitute y = 11/5 into x = 2y - 2:
x = 2(11/5) - 2 = 22/5 - 10/5 = 12/5

Hence, (x, y) = (12/5, 11/5).

Solution

Notice the second equation is just 2 times the first. If we multiply x + 2y = 1 by 2, we get 2x + 4y = 2, which is the same as eqn. 2. This means the two equations are dependent, representing the same line.

There are infinitely many solutions. All points (x, y) satisfying x + 2y = 1 are on this line.

Solution

Observe that equation 1 can be simplified by dividing all terms by 3:
3x + 9y = 6 => x + 3y = 2

This is exactly the same as equation 2. So again, we have infinitely many solutions.

Solution

The left sides of the two equations are identical (x + y), but the right sides differ (10 vs. 7). This implies no solution exists because x + y can’t simultaneously be 10 and 7.

Therefore, the system is inconsistent, and there is no solution.

Solution

Try elimination. Multiply eqn. 1 by 2, eqn. 2 by 3 to align coefficients:
Eqn. 1 * 2: 4x - 6y = 2
Eqn. 2 * 3: 9x + 6y = 51

Now add these two new equations:
(4x - 6y) + (9x + 6y) = 2 + 51
13x = 53
x = 53/13 = 4.0769... but as a fraction, x = 53/13 = 4 + 1/13

Now substitute x = 53/13 into 2x - 3y = 1:
2(53/13) - 3y = 1
106/13 - 3y = 1

Convert 1 to a fraction with denominator 13: 1 = 13/13
106/13 - 3y = 13/13

Subtract 106/13 from both sides:
-3y = 13/13 - 106/13 = -93/13

Divide by -3:
y = (-93/13) / -3 = 93/13 * 1/3 = 93/39 = 3 * 31 / 3 * 13 = 31/13

So, (x, y) = (53/13, 31/13).

Solution

Add the two equations to eliminate y:
(x - y) + (2x + y) = 6 + 9
3x = 15
x = 5

Substitute x = 5 into x - y = 6:
5 - y = 6
-y = 1
y = -1

Hence, (x, y) = (5, -1).

Solution

Add the equations to eliminate y:
(2x - y) + (3x + y) = 3 + 7
5x = 10
x = 2

Substitute x = 2 into 2x - y = 3:
2(2) - y = 3
4 - y = 3
-y = -1
y = 1

The solution is (x, y) = (2, 1).

Solution

Since both expressions equal y, set them equal:
-3x + 12 = x - 8

Add 3x to both sides:
12 = 4x - 8

Add 8 to both sides:
20 = 4x

x = 20/4 = 5

Substitute x = 5 back into y = x - 8:
y = 5 - 8 = -3

Hence, (x, y) = (5, -3).