Simplifying and Combining Like Terms: Solving an equation using all techniques

To solve the equation $-4x = 1 + x$, let's follow these steps:

• Step 1: Subtract $x$ from both sides of the equation to start isolating $x$. This gives: $-4x - x = 1 + x - x$.
• Step 2: Simplify the equation. Combine like terms: $-5x = 1$.
• Step 3: Divide both sides by $-5$ to solve for $x$: $x = \frac{1}{-5}$.

Therefore, the solution to the equation is $x = -\frac{1}{5}$.

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 $-7x + 3 - \frac{1}{2} = 0$, follow these steps:

Step 1: Simplify the equation.
First, combine the constant terms $3$ and $-\frac{1}{2}$. Convert $3$ to a fraction as $\frac{6}{2}$ to facilitate subtraction. Thus, $3 - \frac{1}{2} = \frac{6}{2} - \frac{1}{2} = \frac{5}{2}$.
The equation now becomes: $-7x + \frac{5}{2} = 0$.

Step 2: Move the constant term to the other side.
Subtract $\frac{5}{2}$ from both sides:
$-7x + \frac{5}{2} - \frac{5}{2} = 0 - \frac{5}{2}$.
This simplifies to: $-7x = -\frac{5}{2}$.

Step 3: Isolate $x$.
Divide both sides by $-7$ to solve for $x$:
$x = \frac{-\frac{5}{2}}{-7}$.
Simplify the expression:
$x = \frac{5}{14}$.

Thus, the solution to the equation is $\boxed{\frac{5}{14}}$, which corresponds to choice 3.

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

• Step 1: Subtract $3x$ from both sides of the equation.
• Step 2: Simplify and solve for $x$.

Now, let's work through each step:

Step 1: Subtract $3x$ from both sides
Starting with the equation $5x = \frac{1}{2} + 3x$, subtract $3x$ from both sides to get:
$(5x - 3x) = \frac{1}{2}$.

Step 2: Simplify and solve for $x$
Simplifying the left side yields $2x = \frac{1}{2}$.
Next, divide both sides of the equation by 2 to isolate $x$:
$x = \frac{1}{2} \div 2$.

When dividing $\frac{1}{2}$ by 2, we are effectively finding half of $\frac{1}{2}$, which is:
$x = \frac{1}{4}$.

Therefore, the solution to the problem is $x = \frac{1}{4}$.

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 $b = 1$.

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 $12y + 3y - 10 + 7(y - 4) = 2y$, 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 $y = 1.9$.

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 $x = -3$.

To solve the equation $-x + 3(x-4) = 5 - \frac{1}{2}x$, follow these steps:

• **Step 1**: Distribute the $3$ on the left side:
  $-x + 3x - 12 = 5 - \frac{1}{2}x$.
• **Step 2**: Combine like terms on the left:
  $(2x - 12) = 5 - \frac{1}{2}x$.
• **Step 3**: Add $\frac{1}{2}x$ to both sides to get only the variable terms on one side:
  $2x + \frac{1}{2}x - 12 = 5$.
• **Step 4**: Simplify by combining the $x$ terms:
  $\frac{5}{2}x - 12 = 5$.
• **Step 5**: Add 12 to both sides to isolate terms with $x$:
  $\frac{5}{2}x = 17$.
• **Step 6**: Multiply both sides by $\frac{2}{5}$ to solve for $x$:
  $x = 17 \times \frac{2}{5} = \frac{34}{5}$.

Therefore, the solution to the problem is $x = \frac{34}{5}$.

To solve the given equation, $\frac{1}{8}(x - 3) + 5x = 1$, we will proceed with the following steps:

• Step 1: Distribute $\frac{1}{8}$ across the terms inside the parenthesis.

• Step 2: Combine like terms to simplify the equation.

• Step 3: Isolate the variable $x$ to find its value.

Now, let's work through each step:
Step 1: Distribute $\frac{1}{8}$ into the terms inside the parentheses:

$\frac{1}{8} \cdot x - \frac{1}{8} \cdot 3 = \frac{x}{8} - \frac{3}{8}$

So the equation becomes:

$\frac{x}{8} - \frac{3}{8} + 5x = 1$

Step 2: Combine like terms. First, combine the terms with $x$:

$5x + \frac{x}{8} = \frac{40x}{8} + \frac{x}{8} = \frac{41x}{8}$

Substitute back into the equation:

$\frac{41x}{8} - \frac{3}{8} = 1$

Step 3: Isolate $x$: add $\frac{3}{8}$ to both sides:

$\frac{41x}{8} = 1 + \frac{3}{8}$$\frac{41x}{8} = \frac{8}{8} + \frac{3}{8} = \frac{11}{8}$

Multiply both sides by 8 to eliminate the fraction:

$41x = 11$

Finally, divide both sides by 41 to solve for $x$:

$x=\frac{11}{41}$

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 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 linear equation, we will follow these steps:

• Simplify the left-hand side by expanding and combining like terms.
• Isolate $x$ on one side of the equation.
• Solve for $x$.

Let's perform these steps:

We start with the equation: $-x + 8 \cdot \frac{1}{3}x + 5 = 1 - x$.

First, simplify the term $8 \cdot \frac{1}{3}x$ to $\frac{8}{3}x$.

The equation becomes:

$-x + \frac{8}{3}x + 5 = 1 - x$.

Combine the like terms $-x$ and $\frac{8}{3}x$:

$\left(-1 + \frac{8}{3}\right)x = \frac{8}{3}x - \frac{3}{3}x = \frac{5}{3}x$.

The equation simplifies to:

$\frac{5}{3}x + 5 = 1 - x$.

Add $x$ to both sides to eliminate $x$ on the right-hand side:

$\frac{5}{3}x + x + 5 = 1$.

Convert $x$ to a fraction: $x = \frac{3}{3}x$, so:

$\left(\frac{5}{3} + \frac{3}{3}\right)x + 5 = 1$.

This simplifies to:

$\frac{8}{3}x + 5 = 1$.

Subtract $5$ from both sides to isolate terms involving $x$:

$\frac{8}{3}x = 1 - 5$, which simplifies to:

$\frac{8}{3}x = -4$.

Isolate $x$ by multiplying both sides by the reciprocal of $\frac{8}{3}$:

$x = -4 \times \frac{3}{8}$.

Calculate the value:

$x = -\frac{12}{8}$.

Simplify the fraction:

$x = -\frac{3}{2}$.

Therefore, the solution to the equation is $x = -\frac{3}{2}$.

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: $-18$.

We will begin by simplifying both sides of the given equation:

Given: $-8x + \frac{1}{4} - 3 = 0 - 8x$.

• Simplify the left side: Combine the constant terms.

We have $-8x + \frac{1}{4} - 3$. Converting $-3$ to quarters, we get $-3 = -\frac{12}{4}$. Thus, combine:

$-8x + \frac{1}{4} - \frac{12}{4} = -8x - \frac{11}{4}$.

• The right side remains as $0 - 8x = -8x$.
• Now, the equation is:

$-8x - \frac{11}{4} = -8x$.

• Next, subtract $-8x$ from both sides to simplify:

$-\frac{11}{4} = 0$.

This leads to a contradiction since $-\frac{11}{4}$ is not equal to $0$.

Therefore, the equation has no solution for $x$.

Hence, the correct answer is There is no solution.

To solve the equation $-8x + 3 + \frac{1}{5} = -7x + 5(1-x)$, follow these steps:

• Step 1: Simplify the right-hand side by distributing the 5: $-7x + 5(1-x) = -7x + 5 - 5x$.
• Step 2: This leads to $-8x + 3 + \frac{1}{5} = -12x + 5$.
• Step 3: Combine like terms on both sides. The left simplifies to $-8x + \frac{16}{5}$ and the right side already simplified as stated.
• Step 4: Move all terms involving $x$ to one side, and constants to the other: Add $12x$ to both sides to get $12x - 8x = 5 - \frac{16}{5}$.
• Step 5: Simplify the equation: $4x = \frac{25}{5} - \frac{16}{5} = \frac{9}{5}$.
• Step 6: Solve for $x$ by dividing both sides by 4: $x = \frac{9}{20}$.

Therefore, the solution to the problem is $x = \frac{9}{20}$.

To solve the equation $(x+4)(3x-\frac{1}{4}) = 3(x^2+5)$, follow these steps:

• Step 1: Expand the left side of the equation
  $(x + 4)(3x - \frac{1}{4})$

Using the distributive property:

$x(3x) + x(-\frac{1}{4}) + 4(3x) + 4(-\frac{1}{4})$

$= 3x^2 - \frac{x}{4} + 12x - 1$

• Step 2: Simplify the expanded left side
  Combine like terms:

$3x^2 + \left(12x - \frac{x}{4}\right) - 1$

Convert $\frac{x}{4}$ to a common denominator: $\frac{48x}{4} - \frac{x}{4} = \frac{47x}{4}$

Thus, the left side is: $3x^2 + \frac{47x}{4} - 1$

• Step 3: Simplify the right side
  $3(x^2 + 5)$

$= 3x^2 + 15$

• Step 4: Set the simplified expressions equal and solve for $x$

$3x^2 + \frac{47x}{4} - 1 = 3x^2 + 15$

Subtract $3x^2$ from both sides:

$\frac{47x}{4} - 1 = 15$

Add 1 to both sides:

$\frac{47x}{4} = 16$

Multiply both sides by 4 to clear the fraction:

$47x = 64$

• Step 5: Solve for $x$

$x = \frac{64}{47}$

Express $\frac{64}{47}$ as a mixed number:

$x = 1\frac{17}{47}$

Therefore, the solution to the equation is $x = 1\frac{17}{47}$.

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

• Step 1: Expand and simplify the right-hand side.
• Step 2: Set the equation to zero by moving all terms to one side.
• Step 3: Simplify to obtain a standard quadratic equation.
• Step 4: Use the quadratic formula to find the possible solutions for $x$.

Now, let's work through each step:

Step 1:
Expand the right-hand side:
$(-x + 7)(4x - 9) = -x(4x) - x(-9) + 7(4x) - 7(9)$
= $-4x^2 + 9x + 28x - 63$
Considering both sides: $-4(x^2 + 5) = -4x^2 + 9x + 28x - 63 + 5$.

Step 2:
Simplify further by calculating:
$-4x^2 - 20 = -4x^2 + 37x - 58$.

Step 3:
Move all terms to one side to achieve zero on the right-hand side:
$-4x^2 - 20 + 4x^2 - 37x + 58 = 0$
Simplifying, we get: $37x + 38 = 0$.

Step 4:
Since the $x^2$ terms cancel, it's actually a linear equation:
$37x = -38$.
Solving for $x$, we divide both sides by 37:
$x = \frac{-38}{37} = -1\frac{1}{37}$.

Therefore, the solution to the problem is $x = 1\frac{1}{37}$.