Solving Quadratic Equations using Factoring: Equations with variables on both sides

To open parentheses we will use the formula:

$a(x+b)=ax+ab$

$(7\times-2x)+(7\times5)=77$

We multiply accordingly

$-14x+35=77$

We will move the 35 to the right section and change the sign accordingly:

$-14x=77-35$

We solve the subtraction exercise on the right side and we will obtain:

$-14x=42$

We divide both sections by -14

$\frac{-14x}{-14}=\frac{42}{-14}$

$x=-3$

Let's solve the given linear equation step-by-step:

• Step 1: Distribute the $-6$ across the terms inside the parenthesis:
  The given equation is $(y - 5) - 6(3 + y) = 7$.
  Distribute $-6$ to both $3$ and $y$:
  $-6 \times 3 = -18$ and $-6 \times y = -6y$.
  Rewrite the equation as: $y - 5 - 18 - 6y = 7$.
• Step 2: Combine like terms:
  Combine $y$ and $-6y$ to get $-5y$.
  Combine $-5$ and $-18$ to get $-23$.
  This simplifies the equation to: $-5y - 23 = 7$.
• Step 3: Isolate the variable $y$:
  Add $23$ to both sides to eliminate $-23$:
  $-5y - 23 + 23 = 7 + 23$ simplifies to $-5y = 30$.
  Divide both sides by $-5$ to solve for $y$:
  $y = \frac{30}{-5} = -6$.

Thus, the solution to the equation is $y = -6$.

To solve the given equation $3(b-1)-4(-b+3)=-28$, let's follow these steps:

• Step 1: Distribute the constants.
• Step 2: Simplify by combining like terms.
• Step 3: Isolate the variable $b$.

Now, let's work through each step:
Step 1: Apply the distributive property.
Starting with $3(b-1)-4(-b+3)$, distribute the constants:
$3 \cdot (b) + 3 \cdot (-1) - 4 \cdot (-b) - 4 \cdot 3 = 3b - 3 + 4b - 12$

Step 2: Combine like terms.
Combine the terms involving $b$ and the constant terms:
$3b + 4b - 3 - 12 = 7b - 15$
Set this equal to the right side of the equation:
$7b - 15 = -28$

Step 3: Solve for $b$.
Add 15 to both sides to isolate the term with $b$:
$7b = -28 + 15$

This simplifies to:
$7b = -13$

Finally, divide both sides by 7 to solve for $b$:
$b = \frac{-13}{7}$

Therefore, the solution to the problem is $b = -1\frac{6}{7}$.

Reviewing the answer choices, our solution $b = -1\frac{6}{7}$ matches the correct answer choice.

To solve the problem $2(x-4)+6(x+2)=-18$, we follow these steps:

• Step 1: Apply the distributive property.

Expand both terms using the distributive property:

$2(x-4)$ expands to $2x - 8$ and $6(x+2)$ expands to $6x + 12$.

Thus, the equation becomes:

• Step 2: Combine like terms.

Combine $2x$ and $6x$ to get $8x$, and combine $-8$ and $12$ to get $4$.

The equation simplifies to:

• Step 3: Isolate the variable $x$.

Subtract $4$ from both sides to isolate terms with $x$:

Simplifying the right side gives:

• Step 4: Solve for $x$.

Divide both sides by $8$ to solve for $x$:

Simplify $\frac{-22}{8}$ to $-\frac{11}{4}$, which is $-2\frac{3}{4}$ in mixed number form.

Therefore, the solution to the equation is $x = -2\frac{3}{4}$.

To solve the equation $(x-4)\cdot3=2(x+6)$, we'll follow a systematic approach:

Step 1: Apply the distributive property to both sides of the equation.

• On the left side: $3(x-4) = 3x - 12$
• On the right side: $2(x+6) = 2x + 12$

Step 2: Rewrite the equation with the expanded terms:
$3x - 12 = 2x + 12$

Step 3: Move all terms involving $x$ to one side and constant terms to the other.

• Subtract $2x$ from both sides to move $x$ terms to the left:
  $3x - 2x = 12 + 12$ which simplifies to $x - 12 = 12$.
• Add 12 to both sides to isolate $x$:
  $x = 24$.

Therefore, the solution to the equation is $x = 24$.

We open the parentheses in both sections by the distributive property and use the formula:

$a(x+b)=ax+ab$

$-18+9x=3x+12$

We move 3X to the left section, and 18 to the right section and maintain the corresponding signs:

$9x-3x=12+18$

We add the terms:

$6x=30$

We divide both sections by 6:

$\frac{6x}{6}=\frac{30}{6}$

$x=5$

Let's solve the equation $5(2-x) + 2x = 3(4-x)$ step by step:

First, apply the distributive property to both sides of the equation:

• Left side: $5(2-x) = 5 \cdot 2 - 5 \cdot x = 10 - 5x$.
• Right side: $3(4-x) = 3 \cdot 4 - 3 \cdot x = 12 - 3x$.

Substituting back, the equation becomes:

$10 - 5x + 2x = 12 - 3x$.

Next, combine like terms on the left side:

$10 - 3x = 12 - 3x$.

At this point, notice that the terms involving $x$ on both sides are identical ($-3x$). This means the terms containing $x$ cancel each other out:

$10 = 12$.

Since $10 \neq 12$, we encounter a contradiction.

This implies that there is no solution to the equation. The given equation represents parallel lines that never intersect, so there is no value of $x$ that satisfies the equation.

Therefore, the solution to the problem is There is no solution.

To effectively solve the equation $-2(3x+2)+1=3(x+8)$, we'll break down each step systematically:

• Step 1: Distribute the constants $-2$ and $3$ to the terms within the parentheses on each side of the equation.
• Step 2: Simplify by combining like terms on each side.
• Step 3: Isolate the $x$ term by performing the appropriate arithmetic operations.
• Step 4: Solve for $x$.

Step 1:

Distribute $-2$ in $-2(3x + 2)$:

$-2 \times 3x = -6x$ and $-2 \times 2 = -4$.

Thus, the left side becomes $-6x - 4 + 1$.

Distribute $3$ in $3(x + 8)$:

$3 \times x = 3x$ and $3 \times 8 = 24$.

So, the right side becomes $3x + 24$.

Step 2:

Simplify both sides:

Left side: $-6x - 4 + 1 = -6x - 3$

Right side: $3x + 24$

So the equation becomes:

$-6x - 3 = 3x + 24$

Step 3:

To isolate $x$, add $6x$ to both sides:

$-3 = 9x + 24$

Step 4:

Subtract $24$ from both sides:

$-3 - 24 = 9x$

$-27 = 9x$

Divide both sides by $9$ to solve for $x$:

$x = \frac{-27}{9}$

$x = -3$

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

To solve the equation $-7(x+4)-2=5(2-x)$, follow these steps:

• Step 1: Distribute the coefficients across the terms inside the parentheses.
  On the left side: Apply $-7$ to $(x + 4)$:
  $-7 \cdot x + (-7) \cdot 4 = -7x - 28$.
  On the right side: Apply $5$ to $(2 - x)$:
  $5 \cdot 2 + 5 \cdot (-x) = 10 - 5x$.
• Step 2: Substitute the distributed expressions back into the equation:
  $-7x - 28 - 2 = 10 - 5x$.
• Step 3: Simplify both sides by combining like terms:
  The left side simplifies to: $-7x - 30$.
  Thus, the equation becomes: $-7x - 30 = 10 - 5x$.
• Step 4: Get all terms involving $x$ on one side of the equation:
  Add $5x$ to both sides: $-7x + 5x - 30 = 10$.
  This simplifies to: $-2x - 30 = 10$.
• Step 5: Isolate the term with $x$ on one side:
  Add $30$ to both sides: $-2x = 40$.
• Step 6: Solve for $x$ by dividing by $-2$:
  $x = \frac{40}{-2} = -20$.

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

To open parentheses we will use the formula:

$a\left(x+b\right)=ax+ab$

$(-8\times2x)+(-8\times4)=(6\times x)+(6\times-4)+3$

We multiply accordingly:

$-16x-32=6x-24+3$

Calculate the elements on the right section:

$-16x-32=6x-21$

In the left section we enter the elements with the X and in the left section those without the X, remember to change the plus and minus signs as appropriate when transferring:

$-32+21=6x+16x$

Calculate the elements accordingly

$-11=22x$

We divide the two sections by 22

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

$-\frac{1}{2}=x$

To solve this problem, let's break it down step-by-step:

• Step 1: Expand the equation using the distributive property.

Apply the distributive property to each term: $-2(4 - 3x) + 4(2x - 4) = 8(2 - x)$ $-2 \cdot 4 + (-2) \cdot (-3x) + 4 \cdot 2x - 4 \cdot 4 = 8 \cdot 2 - 8 \cdot x$ which simplifies to:

$-8 + 6x + 8x - 16 = 16 - 8x$

• Step 2: Gather like terms.

Combine $x$ terms and constants separately: $6x + 8x = 14x \quad \text{and} \quad -8 - 16 = -24$ Thus, the equation becomes: $14x - 24 = 16 - 8x$

• Step 3: Isolate $x$ on one side.

Start by adding $8x$ to both sides to bring all terms involving $x$ to one side: $14x + 8x - 24 = 16$ which simplifies to: $22x - 24 = 16$

• Step 4: Solve for $x$.

Add 24 to both sides to isolate the term with $x$: $22x = 16 + 24$ $22x = 40$ Finally, divide both sides by 22: $x = \frac{40}{22} = \frac{20}{11}$

Therefore, the solution to the equation is $\frac{20}{11}$.

To open parentheses we will use the formula:

$a\left(x+b\right)=ax+ab$

$(-7\times2x)+(-7\times3)+(-4\times x)+(-4\times2)=(5\times2)+(5\times-3x)$

We multiply accordingly:

$-14x-21-4x-8=10-15x$

We calculate the elements in the left section:

$-18x-29=10-15x$

In the left section we enter the elements with the X and in the right section those without the X, remember to change the plus and minus signs as appropriate when transferring:

$-18x+15x=10+29$

We calculate the elements accordingly:

$-3x=39$

We divide the two sections by -3:

$\frac{-3x}{-3}=\frac{39}{-3}$

$x=-13$

To solve this linear equation, we will proceed step by step:

• Step 1: Apply the distributive property to both sides of the given equation.
• Step 2: Combine like terms on both sides.
• Step 3: Rearrange the equation to isolate the variable $x$.
• Step 4: Solve for $x$ and verify against the choices provided.

Now, let's work through each step:

Step 1: Apply the distributive property:
$-8(4 - x)$ becomes $-32 + 8x$ and $4(2x + 5)$ becomes $8x + 20$.
The right side $2(7 - 2x)$ becomes $14 - 4x$.

This gives us the new equation:
$-32 + 8x + 8x + 20 = 14 - 4x$.

Step 2: Combine like terms:
On the left side: $-32 + 20 + 8x + 8x = -12 + 16x$.
On the right side: $14 - 4x$ remains unchanged.

The equation simplifies to:
$-12 + 16x = 14 - 4x$.

Step 3: Rearrange the equation to isolate $x$.
Add $4x$ to both sides to move the $x$ terms to one side:
$-12 + 16x + 4x = 14$.
This simplifies to:
$-12 + 20x = 14$.

Next, add 12 to both sides to isolate terms with $x$:
$20x = 14 + 12$.
Thus, $20x = 26$.

Step 4: Solve for $x$:
Divide by 20:
$x = \frac{26}{20} = \frac{13}{10}$.

Therefore, the correct solution is $x = \frac{13}{10}$, which corresponds to choice 3.

To solve this equation $-2(4 + 5x) + 3(2 - 2x) = 8(4 - x)$, let's work through it step by step:

Step 1: Distribute the constants into the parentheses:

• Distribute $-2$ in $-2(4 + 5x)$:
$-2(4) + -2(5x) = -8 - 10x$
• Distribute $3$ in $3(2 - 2x)$:
$3(2) + 3(-2x) = 6 - 6x$
• Distribute $8$ in $8(4 - x)$:
$8(4) - 8(x) = 32 - 8x$

Step 2: Substitute the distributed expressions back into the equation:

Step 3: Combine like terms on each side of the equation:

• On the left side, combine $-8$ and $6$, and $-10x$ and $-6x$:
$-2 - 16x$
• The right side remains:
$32 - 8x$

Step 4: Rewrite the equation:

Step 5: Isolate $x$ by first adding $8x$ to both sides to get rid of $-8x$ on the right:

Step 6: Add $2$ to both sides to isolate the x term:

Step 7: Divide both sides by $-8$ to solve for $x$:

Therefore, the solution to the equation is $x = -\frac{17}{4}$.

To solve the equation $(\frac{1}{2}x + 3) - (4x + 7) = 1$, follow these steps:

• Simplify the Left Side:

Begin by distributing the negative sign to the terms in the parentheses on the left-hand side: $(\frac{1}{2}x + 3) - 4x - 7 = 1$

Combine like terms: $\frac{1}{2}x - 4x + 3 - 7 = 1$

This simplifies to: $-\frac{7}{2}x - 4 = 1$

• Isolate the Variable $x$:

Add 4 to both sides to isolate the term involving $x$: $-\frac{7}{2}x - 4 + 4 = 1 + 4$

Resulting in: $-\frac{7}{2}x = 5$

• Solve for $x$:

Multiply both sides by the reciprocal of $-\frac{7}{2}$, which is $-\frac{2}{7}$, to solve for $x$: $x = 5 \times \left(-\frac{2}{7}\right)$

Calculate the product: $x = -\frac{10}{7}$

Convert to a mixed number: $x = -1\frac{3}{7}$

Therefore, the solution to the problem is $x = -1\frac{3}{7}$.

The correct choice from the options given is choice $-1\frac{3}{7}$ (Choice 3).