Solving Equations by using Addition/ Subtraction: Solving an equation using all techniques

We begin by simplifying the given equation:

$37b + 6b + 56 = 90 + 9$

First, we combine like terms on the left side of the equation:

$(37 + 6)b + 56$

This simplifies to:

$43b + 56$

Now the equation is:

$43b + 56 = 99$

Next, we need to isolate $b$ by moving the constant term to the right side. We do this by subtracting 56 from both sides:

$43b = 99 - 56$

Simplifying the right-hand side gives us:

$43b = 43$

Finally, to solve for $b$, we divide both sides by 43:

$b = \frac{43}{43}$

This simplifies to:

$b = 1$

Therefore, the solution to the problem is $b = 1$.

To solve the equation $7y + 10y + 5 = 2(y + 3)$, let's proceed as follows:

• Step 1: Simplify the left side by combining like terms. The expression $7y + 10y$ combines to $17y$, so we have $17y + 5 = 2(y + 3)$.

• Step 2: Expand the right side. Distribute the 2 across the parenthesis: $2(y + 3)$ becomes $2y + 6$. The equation now reads $17y + 5 = 2y + 6$.

• Step 3: Isolate terms involving $y$ on one side. Subtract $2y$ from both sides: $17y - 2y + 5 = 6$, which simplifies to $15y + 5 = 6$.

• Step 4: Isolate $15y$ by subtracting 5 from both sides: $15y = 6 - 5$, which simplifies to $15y = 1$.

• Step 5: Solve for $y$ by dividing both sides by 15: $y = \frac{1}{15}$.

Therefore, the solution to the problem is $\mathbf{y = \frac{1}{15}}$.

To solve the equation $6c + 7 + 4c = 3(c - 1)$, follow these steps:

• Step 1: Combine like terms on the left side of the equation.
  The like terms are $6c$ and $4c$. Combining these gives $10c + 7 = 3(c - 1)$.
• Step 2: Apply the distributive property on the right side of the equation.
  The term $3(c - 1)$ expands to $3c - 3$. Therefore, the equation becomes $10c + 7 = 3c - 3$.
• Step 3: Move all terms involving $c$ to one side and constants to the other.
  Subtract $3c$ from both sides: $10c - 3c + 7 = -3$ which simplifies to $7c + 7 = -3$.
• Step 4: Isolate the term with $c$ by subtracting 7 from both sides of the equation.
  This gives $7c = -3 - 7$ or $7c = -10$.
• Step 5: Solve for $c$.
  Divide both sides by 7: $c = \frac{-10}{7} = -\frac{10}{7}$. This can be converted to a mixed number, giving $-1\frac{3}{7}$.

Therefore, the solution to the equation is $c = -1\frac{3}{7}$. This corresponds to choice 2 in the provided answer choices.

To solve the equation $4y - 7 + 6y = 3 - 10y$, follow these steps:

• Step 1: Combine like terms on both sides of the equation.

The left side simplifies by combining the $y$-terms:
$4y + 6y - 7 = 10y - 7$.

On the right side, there is one $y$-term, but we can leave it for the next steps.

• Step 2: Isolate the $y$-terms.

Add $10y$ to both sides to move all $y$-terms to the left side:

$10y - 7 + 10y = 3$

Simplifying the left side, we get:

$20y - 7 = 3$

• Step 3: Isolate $y$.

Add $7$ to both sides to eliminate the constant term on the left:

$20y = 3 + 7$

$20y = 10$

Divide both sides by $20$ to solve for $y$:

$y = \frac{10}{20}$

$y = \frac{1}{2}$

Therefore, the solution to the problem is $y = \frac{1}{2}$.

To solve the equation $\frac{1}{4}a + 5 = 20 + a$, follow these steps:

• Step 1: Eliminate the $a$ variable term on the right side:
  Subtract $a$ from both sides:
  $\frac{1}{4}a + 5 - a = 20 + a - a$
  This simplifies to:
  $\frac{1}{4}a - a + 5 = 20$.
• Step 2: Combine like terms involving $a$:
  The equation becomes: $\frac{1}{4}a - \frac{4}{4}a + 5 = 20$
  Combine them to get:
  $-\frac{3}{4}a + 5 = 20$.
• Step 3: Isolate the $a$ term:
  Subtract 5 from both sides:
  $-\frac{3}{4}a + 5 - 5 = 20 - 5$
  This simplifies to:
  $-\frac{3}{4}a = 15$.
• Step 4: Solve for $a$:
  To isolate $a$, divide both sides by $-\frac{3}{4}$:
  $a = \frac{15}{-\frac{3}{4}} = 15 \times -\frac{4}{3} = -20$.
  Therefore, $a = -20$.

Therefore, the solution to the equation is $a = -20$.

To solve the given equation $8 + 2(x + 1) = 7x$, we'll follow these steps:

• Step 1: Distribute the $2$ inside the parentheses.
• Step 2: Combine like terms and simplify both sides.
• Step 3: Isolate the variable $x$ on one side of the equation.

Now, let's apply these steps:
Step 1: Distribute the $2$ in the equation: $2(x + 1) = 2x + 2$ Therefore, the equation becomes: $8 + 2x + 2 = 7x$ Step 2: Combine like terms on the left side: $10 + 2x = 7x$ Step 3: Isolate $x$ by moving all terms involving $x$ to one side and constants to the other: $10 = 7x - 2x$ $10 = 5x$ To solve for $x$, divide both sides by $5$: $x = \frac{10}{5} = 2$

Therefore, the solution to the equation is $x = 2$.

To solve the equation $3 - (x - 4) + 8x = 14$, we will simplify and solve for $x$ step-by-step:

• Step 1: Distribute the negative sign inside the parentheses:
  $- (x - 4) = -x + 4$.

• Step 2: Rewrite the equation with the distributed terms:
  $3 + (-x + 4) + 8x = 14$.

• Step 3: Simplify by combining like terms:
  Combine the constants: $3 + 4 = 7$
  Combine the $x$ terms: $-x + 8x = 7x$.
  This gives us $7 + 7x = 14$.

• Step 4: Isolate the $x$ term:
  Subtract 7 from both sides: $7x = 14 - 7$.
  This simplifies to $7x = 7$.

• Step 5: Solve for $x$ by dividing both sides by 7:
  $x = \frac{7}{7}$.
  Simplifying gives $x = 1$.

Therefore, the solution to the equation is $x = 1$.

To solve the equation $12y + 3y - 10 + 7(y - 4) = 2y$, follow these detailed steps:

• Step 1: Apply the distributive property to $7(y - 4)$.

This results in:
$12y + 3y - 10 + 7y - 28 = 2y$.

• Step 2: Combine like terms on the left side of the equation.

Combining terms, we have:
$(12y + 3y + 7y) - 10 - 28 = 2y$
$22y - 38 = 2y$.

• Step 3: Move all terms involving $y$ to one side of the equation and constant terms to the other side.

Subtract $2y$ from both sides:
$22y - 2y = 38$
$20y = 38$.

• Step 4: Solve for $y$ by dividing both sides by 20.

$y = \frac{38}{20} = 1.9$.

Therefore, the solution to the equation is $y = 1.9$.

Let's solve the problem step-by-step:

Step 1: Begin with the original equation:
$16a - 20a + 15 = 2(5 - 2a)$.

Step 2: Simplify the left-hand side by combining like terms:
$16a - 20a + 15 = -4a + 15$.

Step 3: Apply the distributive property to the right-hand side:
$2(5 - 2a) = 2 \cdot 5 - 2 \cdot 2a = 10 - 4a$.

Step 4: Now the equation reads:
$-4a + 15 = 10 - 4a$.

Step 5: Attempt to isolate the variable $a$ by subtracting $10$ from both sides:
$-4a + 15 - 10 = -4a$.
This simplifies to:
$-4a + 5 = -4a$.

Step 6: Subtract $-4a$ from both sides to further simplify the equation:
$5 = 0$.
This is a contradiction, indicating that no solution exists for the equation since a statement like this is never true.

Therefore, the solution to the problem is No solution.

To solve the equation $3x + \frac{2}{3} = 4 \left(x + \frac{1}{12}\right)$, we follow these steps:

Step 1: Distribute the 4 on the right-hand side.

$4(x + \frac{1}{12}) = 4x + \frac{4}{12}$ which simplifies to $4x + \frac{1}{3}$.

Step 2: Write down the modified equation.

The equation now reads: $3x + \frac{2}{3} = 4x + \frac{1}{3}$.

Step 3: Rearrange the equation to collect like terms.

Subtract $3x$ from both sides: $3x + \frac{2}{3} - 3x = 4x + \frac{1}{3} - 3x$.

This simplifies to: $\frac{2}{3} = x + \frac{1}{3}$.

Step 4: Isolate $x$.

Subtract $\frac{1}{3}$ from both sides: $\frac{2}{3} - \frac{1}{3} = x$.

This simplifies to: $x = \frac{1}{3}$.

Therefore, the solution to the equation is $\boxed{\frac{1}{3}}$.

To solve the equation $\frac{1}{3}(x+9) = 4+\frac{2}{3}x$, we will follow these steps:

• Step 1: Clear fractions by multiplying through by the least common denominator.
• Step 2: Simplify the equation to combine like terms.
• Step 3: Solve for $x$.

Let's begin:

Step 1: Multiply every term in the equation by 3 to eliminate fractions:

This simplifies to:

Step 2: Rearrange the equation to get all $x$ terms on one side and constant terms on the other:

Subtract $2x$ from both sides:

Which simplifies to:

Next, subtract 9 from both sides to isolate terms involving $x$:

Step 3: Solve for $x$ by multiplying both sides by -1:

Thus, the solution to the equation is $x = -3$.

First, let's isolate a from the parentheses in the equation on the right side. We'll remember that minus times minus becomes plus, so we get the equation:

$a^4+7a-5=2a+a^4+3a+a$

Let's continue solving the equation on the right side by adding $2a+3a+a=5a+a=6a$

Now the equation we got is:

$a^4+7a-5=6a+a^4$

Let's divide both sides by $a^4$ and we get:

$7a-5=6a$

Now let's move 6a to the left side and the number 5 to the right side, remembering to change the plus and minus signs accordingly.

The equation we got now is:

$7a-6a=5$

Let's solve the subtraction and we get:

$1a=5$

Let's divide both sides by 1 and we find that $a=5$

To solve the equation $6\cdot(5+2x)=3x-6$, we will proceed as follows:

• Step 1: Distribute the 6 within the parentheses on the left side:
  $6\cdot(5+2x) = 30 + 12x$.
• Step 2: Substitute the distributed expression into the equation:
  $30 + 12x = 3x - 6$.
• Step 3: Isolate the variable terms on one side:
  Subtract $3x$ from both sides: $30 + 12x - 3x = -6$, which simplifies to $30 + 9x = -6$.
• Step 4: Isolate the term containing $x$:
  Subtract 30 from both sides: $9x = -6 - 30$, which simplifies to $9x = -36$.
• Step 5: Solve for $x$ by dividing both sides by 9:
  $x = \frac{-36}{9} = -4$.

Therefore, the solution to the equation is $x = -4$, which corresponds to choice 1.

To open the parentheses on the left side, we'll use the formula:

$-a\left(b+c\right)=-ab-ac$

$-12a-24=27a$

We'll arrange the equation so that the terms with 'a' are on the right side, and maintain the plus and minus signs during the transfer:

$-24=27a+12a$

Let's group the terms on the right side:

$-24=39a$

Let's divide both sides by 39:

$-\frac{24}{39}=\frac{39a}{39}$

$-\frac{24}{39}=a$

Note that we can reduce the fraction since both numerator and denominator are divisible by 3:

$-\frac{8}{13}=a$

To solve this problem, we'll follow these steps:

• Step 1: Distribute and simplify the equation.
• Step 2: Combine like terms to isolate $x$ on one side of the equation.
• Step 3: Solve for $x$.

Let's begin solving the equation:

Step 1: Distribute and simplify.
The given equation is $(3 - 2x) \cdot 5 = 4 + 12x$.
First, we distribute 5 to both terms inside the parenthesis:

$5 \cdot 3 - 5 \cdot 2x = 4 + 12x$.

This simplifies to:

$15 - 10x = 4 + 12x$.

Step 2: Combine like terms to isolate $x$ on one side of the equation.
Add $10x$ to both sides to get all terms involving $x$ on the right-hand side:

$15 = 4 + 12x + 10x$.

This simplifies to:

$15 = 4 + 22x$.

Subtract 4 from both sides to isolate the term involving $x$:

$15 - 4 = 22x$.

This simplifies to:

$11 = 22x$.

Step 3: Solve for $x$.
To find $x$, divide both sides of the equation by 22:

$x = \frac{11}{22}$.

Upon simplification, $\frac{11}{22}$ reduces to $\frac{1}{2}$.

Therefore, the solution to the equation is $x = \frac{1}{2}$.

To solve the given linear equation $-\frac{7}{4}(-x) + 2x - 5(x + 3) = -x$, follow these steps:

• Step 1: Distribute the coefficients across the terms within parentheses:
  The term $-\frac{7}{4}(-x)$ becomes $\frac{7}{4}x$ because $-\frac{7}{4} \times -x = \frac{7}{4}x$.
  The term $-5(x + 3)$ can be expanded to $-5x - 15$.
• Step 2: Simplify the equation by combining like terms:
  The equation becomes $\frac{7}{4}x + 2x - 5x - 15 = -x$.
• Step 3: Combine the $x$-terms on the left side:
  Combine: $\frac{7}{4}x + 2x - 5x$.
  Converting all terms to a common denominator, $2x = \frac{8}{4}x$ and $-5x = \frac{-20}{4}x$. Thus,
  $\frac{7}{4}x + \frac{8}{4}x - \frac{20}{4}x = \frac{-5}{4}x$.
• Step 4: The equation simplifies to:
  $\frac{-5}{4}x - 15 = -x$.
• Step 5: Isolate the $x$ terms onto one side:
  Add $x$ to both sides, treating $-x$ as $\frac{-4}{4}x$:
  $\frac{-5}{4}x + x - 15 = 0$, which simplifies to $\frac{-1}{4}x - 15 = 0$.
• Step 6: Isolate $x$:
  Add $15$ to both sides:
  $\frac{-1}{4}x = 15$.
• Step 7: Solve for $x$:
  Multiply both sides by $-4$ to isolate $x$:
  $x = 15 \times -4$.
  Thus, $x = -60$.

Therefore, the solution to the equation is $x = -60$.

To solve this problem, we'll follow these steps:

• Step 1: Expand the left-hand side of the equation.
• Step 2: Set the equation to standard quadratic form.
• Step 3: Factor the quadratic equation.
• Step 4: Solve for $x$.

Let's proceed through each step:

Step 1: Expand the left-hand side using the distributive property:

$(x+2)(2x-4) = x(2x) + x(-4) + 2(2x) + 2(-4)$

$= 2x^2 - 4x + 4x - 8$

$= 2x^2 - 8$

Step 2: Set the equation to quadratic form:

Set the expanded result equal to the right-hand side:

$2x^2 - 8 = 2x^2 + x + 10$

Step 3: Subtract the right-hand side from the left:

$2x^2 - 8 - (2x^2 + x + 10) = 0$

Simplify:

$2x^2 - 8 - 2x^2 - x - 10 = 0$

$-x - 18 = 0$

Step 4: Solve for $x$:

$-x = 18$

Divide by -1:

$x = -18$

Therefore, the solution to the problem is $x = -18$.

Checking against the given choices, choice 1 matches: $-18$.

First, we'll expand the parentheses by multiplying each term by 4:

$4\times\frac{b}{2}+4\times b-\frac{1}{3}=6b$

Let's then solve the multiplication exercise $4\times\frac{b}{2}=\frac{4b}{2}=2b$.

Now the equation is:

$2b+4b-\frac{1}{3}=6b$

We can now combine the left-hand side between the two $b$ terms to get:

$6b-\frac{1}{3}=6b$

We'll reduce both sides by $6b$, leaving us with:

$-\frac{1}{3}=0$

Since the result obtained is impossible, the exercise has no solution.

To solve this problem, we'll follow these steps:

• Step 1: Distribute on the right-hand side
• Step 2: Combine like terms on the left-hand side
• Step 3: Isolate the variable $x$
• Step 4: Solve for $x$

Now, let's work through each step:

Step 1: Distribute on the right-hand side of the equation:

$2x + 45 - \frac{1}{3}x = 5(x + 7) \quad \Rightarrow \quad 2x + 45 - \frac{1}{3}x = 5x + 35$

Step 2: Combine like terms on the left-hand side:

Combine $2x$ and $-\frac{1}{3}x$ on the left:

$2x - \frac{1}{3}x = \frac{6}{3}x - \frac{1}{3}x = \frac{5}{3}x$

The equation becomes:

$\frac{5}{3}x + 45 = 5x + 35$

Step 3: Move all terms with $x$ to one side and constants to the other:

$\frac{5}{3}x - 5x = 35 - 45$

Step 4: Simplify and solve for $x$:

$\frac{5}{3}x - \frac{15}{3}x = -10$ $-\frac{10}{3}x = -10$

Step 5: Solve for $x$ by dividing both sides by $-\frac{10}{3}$:

$x = \frac{-10}{-\frac{10}{3}} = 3$

Therefore, the solution to the problem is $x = 3$.

To solve this problem, we'll eliminate fractions by multiplying the entire equation by the least common denominator, and then proceed with algebraic manipulation:

• Step 1: Multiply every term by the least common denominator, $4$, to eliminate fractions:
  $4 \times \left(\frac{1}{4}(x - 16) + \frac{3}{4}x\right) = 4 \times (8x - 11)$.

• Step 2: This simplifies to: $1(x - 16) + 3x = 32x - 44$.

• Step 3: Distribute within the brackets:
  $x - 16 + 3x = 32x - 44$.

• Step 4: Combine like terms on the left-hand side:
  $4x - 16 = 32x - 44$.

• Step 5: To isolate terms containing $x$, subtract $4x$ from both sides:
  $-16 = 28x - 44$.

• Step 6: Add 44 to both sides to further isolate $x$:
  $28 = 28x$.

• Step 7: Divide both sides by 28 to solve for $x$:
  $x = 1$.

Therefore, the solution to the problem is $x = 1$.