Algebraic Solution: Solve using the substitution method

To solve the system of equations using substitution, follow these steps:

• The system of equations given is: $\begin{cases} x + y = 18 \\ y = 13 \end{cases}$
• Step 1: Extract the given value for $y$ from the second equation: $y = 13$.
• Step 2: Substitute $y = 13$ into the first equation: $x + 13 = 18$
• Step 3: Solve for $x$ by subtracting $13$ from both sides of the equation: $x = 18 - 13$
• Step 4: After the subtraction, we find: $x = 5$

Therefore, the solution to the problem is $x = 5$ and $y = 13$.

To solve this system of equations, we'll use the substitution method as follows:

• Step 1: Identify the given information.
  We have two equations: $\begin{cases} 2x + y = 9 \\ x = 5 \end{cases}$
• Step 2: Substitute $x = 5$ into the first equation.
  The equation becomes: $2(5) + y = 9$ which simplifies to: $10 + y = 9$
• Step 3: Solve for $y$.
  Subtract 10 from both sides: $y = 9 - 10$ $y = -1$
• Step 4: Verify the solution.
  Substituting $x = 5$ and $y = -1$ back into the first equation confirms the solution:
  $2(5) + (-1) = 10 - 1 = 9$

Both equations are satisfied with $x = 5$ and $y = -1$.

Therefore, the solution to the system of equations is $x = 5, y = -1$.

To solve this system using the substitution method, we'll follow these steps:

• Step 1: Solve the first equation for one variable.

• Step 2: Substitute this expression into the second equation.

• Step 3: Solve for the second variable.

• Step 4: Use the value of the second variable to find the first variable.

Step 1: Solve the first equation $x + y = 5$ for $y$.
We have: $y = 5 - x$.

Step 2: Substitute $y = 5 - x$ into the second equation $2x - 3y = -15$.
This gives us: $2x - 3(5 - x) = -15$.

Step 3: Simplify and solve for $x$:
$\begin{aligned} 2x - 15 + 3x &= -15 \\ 5x - 15 &= -15 \\ 5x &= 0 \\ x &= 0. \end{aligned}$

Step 4: Substitute $x = 0$ back into $y = 5 - x$ to find $y$.
$\begin{aligned} y &= 5 - 0 \\ y &= 5. \end{aligned}$

Thus, the solution to the system of equations is $x = 0$ and $y = 5$.

The correct answer from the list of choices is: $x = 0, y = 5$

Let's begin by solving the system of equations using the substitution method.

First, solve the second equation for $y$:

$3x + y = 8$

Solve for $y$:

$y = 8 - 3x$

Next, substitute this expression for $y$ in the first equation:

$-x - 2(8 - 3x) = 4$

Distribute the $-2$:

$-x - 16 + 6x = 4$

Combine like terms:

$5x - 16 = 4$

Add 16 to both sides:

$5x = 20$

Divide by 5:

$x = 4$

Now, substitute $x = 4$ back into $y = 8 - 3x$ to find $y$:

$y = 8 - 3(4)$

$y = 8 - 12$

$y = -4$

Therefore, the solution to the system of equations is $(x, y) = (4, -4)$.

Thus, the values of $x$ and $y$ are $x = 4$ and $y = -4$.

To solve the given system of linear equations using the substitution method, follow these steps:

• Step 1: Solve the second equation for $x$.

From the second equation:

We can solve for $x$ as follows:

• Step 2: Substitute the expression for $x$ into the first equation.

Substitute $x = 16 - 8y$ into the first equation:

Simplify and solve for $y$:

- Distribute $-5$:

- Combine like terms:

- Add 80 to both sides:

- Divide by 49:

• Step 3: Substitute $y = 2$ back into the expression for $x$.

The expression for $x$ is:

- Substitute $y = 2$:

Therefore, the values that satisfy both equations in the system are $x = 0$ and $y = 2$.

To solve this problem, we'll apply the substitution method, following these steps:

• Step 1: Solve the first equation for one of the variables.
• Step 2: Substitute this expression into the second equation and solve for the other variable.
• Step 3: Use the value found in Step 2 in the rearranged first equation to find the remaining variable.

Step-by-Step Solution:

Step 1: By using the first equation, $-4x + 4y = 15$, we can solve for $y$.

Step 1.1: Simplify the equation to solve for $y$ by adding $4x$ to both sides:
$4y = 4x + 15$

Step 1.2: Divide every term by 4:
$y = x + \frac{15}{4}$

Step 2: Substitute the expression for $y$ into the second equation, $2x + 8y = 12$.

Step 2.1: Substitute $y = x + \frac{15}{4}$:
$2x + 8(x + \frac{15}{4}) = 12$

Step 2.2: Simplify and solve for $x$:
$2x + 8x + 30 = 12$

Combine like terms:
$10x + 30 = 12$

Subtract 30 from both sides:
$10x = 12 - 30$

Resulting in:
$10x = -18$

Divide by 10:
$x = -\frac{9}{5}$

Step 3: Substitute $x = -\frac{9}{5}$ back into the expression for $y$:

$y = -\frac{9}{5} + \frac{15}{4}$

Convert fractions to a common denominator, which is 20:
$y = -\frac{36}{20} + \frac{75}{20}$

Solve by combining terms:
$y = \frac{39}{20}$

Thus, the solution to the system is $x = -\frac{9}{5}$ and $y = \frac{39}{20}$.

Therefore, the correct solution is identified as choice 4.

To solve this system of equations using the substitution method, we follow these steps:

• Step 1: Solve for one of the variables in terms of the other using the first equation $-5x + y = 8$.
• We solve for $y$:

$y = 5x + 8$

• Step 2: Substitute the expression for $y$ from Step 1 into the second equation $3x - 2y = 11$.

$3x - 2(5x + 8) = 11$

• Step 3: Simplify and solve for $x$.

Simplify the substitution:

$3x - 10x - 16 = 11$

$-7x - 16 = 11$

Add 16 to both sides:

$-7x = 27$

Divide by -7:

$x = -\frac{27}{7}$

• Step 4: Substitute $x$ back into the expression for $y$ from Step 1.

$y = 5\left(-\frac{27}{7}\right) + 8$

Simplify:

$y = -\frac{135}{7} + \frac{56}{7}$

$y = -\frac{135}{7} + \frac{56}{7} = -\frac{79}{7}$

Therefore, the solution to the system is $x = -\frac{27}{7}$ and $y = -\frac{79}{7}$.

To solve the system of equations, follow the steps below:

• Simplify the second equation: Start with $2x + 2y = 12$. Divide every term by 2 to simplify it to $x + y = 6$.
• Compare the two equations now: $x + y = 15$ and $x + y = 6$.

Consider these equations:
Since both are simplified to the form $x + y = \text{constant}$, they describe two parallel lines, given that they have the same coefficients of $x$ and $y$ but different constants (15 and 6).

Parallel lines never intersect. Thus, there is no solution for this system of equations, as they represent two distinct parallel lines.

Therefore, the correct answer is: No solution.

To solve this system of equations using the substitution method, follow these steps:

• Solve $3x + 3y = 9$ for $y$:

Divide the whole equation by 3 to simplify:
$x + y = 3$

• Express $y$ in terms of $x$:

$y = 3 - x$

• Substitute $y = 3 - x$ into the first equation:

We substitute into $8x - 2y = 10$:
$8x - 2(3 - x) = 10$

Simplify and solve for $x$:

$8x - 6 + 2x = 10$
$10x - 6 = 10$
Add 6 to both sides:
$10x = 16$
Divide by 10:
$x = \frac{16}{10} = \frac{8}{5}$

• Substitute back to find $y$:

Use $y = 3 - x$:
$y = 3 - \frac{8}{5} = \frac{15}{5} - \frac{8}{5} = \frac{7}{5}$

Therefore, the solution to the system is $x = \frac{8}{5}$ and $y = \frac{7}{5}$.

To solve the given system of equations using the substitution method, follow these steps:

• Step A: Simplify and solve the first equation for $x$.
  Given: $x + 4y = 5 - 3y$
  Combine like terms: $x + 4y + 3y = 5$$x + 7y = 5$
  Now solve for $x$: $x = 5 - 7y$

• Step B: Substitute this expression for $x$ in the second equation: $2x + 3y = 6$
  Substitute $x = 5 - 7y$: $2(5 - 7y) + 3y = 6$
  Expand and simplify: $10 - 14y + 3y = 6$$10 - 11y = 6$$-11y = 6 - 10$$-11y = -4$
  Solving for $y$: $y = \frac{-4}{-11} = \frac{4}{11}$

• Step C: Substitute $y = \frac{4}{11}$ back into the equation for $x$: $x = 5 - 7(\frac{4}{11})$$x = 5 - \frac{28}{11}$
  Convert 5 to an equivalent fraction: $x = \frac{55}{11} - \frac{28}{11}$$x = \frac{27}{11}$

The solution to the system of equations is $x = \frac{27}{11}$ and $y = \frac{4}{11}$.

To solve this system of equations, follow these steps:

• Step 1: Simplify the second equation:
  The second equation is $-4x + 6y = 28$. By dividing every term by 2, we get:
  $-2x + 3y = 14$.
• Step 2: Compare the simplified second equation to the first equation:
  Both equations are now $-2x + 3y = 14$.

Since both equations are identical after simplification, this indicates that the system represents the same line.

Therefore, the system has infinite solutions because any point that satisfies one equation will satisfy the other.

Thus, the correct answer is Infinite solutions.

To solve this problem using the substitution method, we'll carefully examine the structure of the given system of equations:

• Given equations:
  1) $-8x + 5y = 10$
  2) $-24x + 15y = 30$

Notice that the second equation is exactly three times the first equation:

This implies the two equations are not independent; rather, they are multiples of each other.

This insight tells us that every solution of the first equation is also a solution of the second equation, which means:

The system has infinitely many solutions.

Given this conclusion, when examining the choices provided, the correct choice is "Infinite solutions."

Therefore, the solution to the system of equations is that it has infinite solutions.

To solve this problem, we'll break it down into clear steps:

• Step 1: Simplify the given equations for easier manipulation.
• Step 2: Use the substitution method to express one variable in terms of the other from one equation.
• Step 3: Insert this expression into the other equation to solve for one variable.
• Step 4: Finally, substitute back to find the second variable.

Now, let's work through each step:

Step 1: Simplify the first equation.
The first equation is: $\frac{x-y}{2} + \frac{-y+x}{3} = -6$
First, let's find a common denominator of 6 for the fractions: $\frac{3(x-y) + 2(-y+x)}{6} = -6$
This simplifies to: $\frac{3x-3y-2y+2x}{6} = -6$
$\frac{5x-5y}{6} = -6$

Step 1 (cont.): Multiply both sides by 6 to get rid of the denominator:
$5x - 5y = -36$
Divide every term by 5 to make it simpler: $x - y = -\frac{36}{5}$ $x = y - \frac{36}{5}$ (Equation 1)

Step 2: Simplify the second equation.
The second equation is: $\frac{-y-x}{5} - (y + x) = 8$ Multiply through by 5 to clear the fraction from the first term: $-y-x - 5(y + x) = 40$ This expands to: $-y-x - 5y - 5x = 40$ Combine like terms: $-6y - 6x = 40$ Divide every term by -6 for simplicity: $y + x = -\frac{40}{6}$ $y + x = -\frac{20}{3}$ (Equation 2)

Step 3: Substitute for $x$ in Equation 2 using Equation 1.
From (Equation 1), $x = y - \frac{36}{5}$. Substitute this into Equation 2: $y + (y - \frac{36}{5}) = -\frac{20}{3}$ This gives: $2y - \frac{36}{5} = -\frac{20}{3}$ Add $\frac{36}{5}$ to both sides to isolate $2y$: $2y = -\frac{20}{3} + \frac{36}{5}$

Convert both terms to a common denominator to add them together. The common denominator of 3 and 5 is 15: $2y = -\frac{100}{15} + \frac{108}{15}$ This simplifies to: $2y = \frac{8}{15}$ Divide both sides by 2 to solve for $y$: $y = \frac{4}{15} \approx 0.266$

Step 4: Solve for $x$ using $y$'s value in Equation 1.
Plug $y = \frac{4}{15}$ back into Equation 1: $x = \frac{4}{15} - \frac{36}{5}$ To add, convert to a common denominator, which can be 15: $x = \frac{4}{15} - \frac{108}{15}$ $x = \frac{4 - 108}{15}$ $x = -\frac{104}{15} \approx -6.933$

Therefore, after solving both variables, the values that satisfy the given system of equations are:
$x \approx -6.933,\ y \approx 0.266$.

To solve the system of equations using substitution:

Step 1: Simplify each equation.

First equation:

Multiply by 10 to eliminate denominators:

Second equation:

Multiply by 24 to eliminate denominators:

Step 2: Solve the first equation for $x$:

Step 3: Substitute Equation (3) into Equation (2):

Clear while multiplying by 12:

Step 4: Substitute $y = -4.65$ back into Equation (3) to find $x$:

Therefore, the solution to the problem is $x = 1.57$, $y = -4.65$.

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

• Step 1: Verify if both equations in the system are proportional.
• Step 2: Determine if they result in a true or false statement when simplified.
• Step 3: Conclude about the solution set based on the results.

Now, let's work through each step:
Step 1: Start with the system of equations:
$\begin{cases} 3x - 4y = 10 \\ 9x - 12y = 15 \end{cases}$

Step 2: Check if the second equation is a multiple of the first equation.
Divide each coefficient of the second equation by 3:
- $(9x) \div 3 = 3x$
- $(-12y) \div 3 = -4y$
- $(15) \div 3 = 5$
Thus, converting: $\begin{cases} 3x - 4y = 10 \\ 3x - 4y = 5 \end{cases}$

Step 3: We notice that while the left sides of both equations are identical, the right sides differ:
This results in a logical contradiction because $10 \neq 5$.
Thus, these lines are parallel and distinct, indicating that the system has no common points of intersection, hence no solution.

Therefore, the correct conclusion for this system of equations is No solution.

To solve this system, we'll first simplify and manipulate the equations to perform substitution. Let's begin:

The first equation is $\frac{2x-3y}{4} + \frac{x-y}{5} = 30$.
To eliminate the fractions, multiply through by 20 (the least common multiple of 4 and 5):

$20 \left( \frac{2x-3y}{4} \right) + 20 \left( \frac{x-y}{5} \right) = 20 \times 30$

Simplifying each term gives:
$5(2x - 3y) + 4(x - y) = 600$

Expanding the terms:

$10x - 15y + 4x - 4y = 600$

Combine like terms:

$14x - 19y = 600$     [Equation (1)]

Next, let's work on the second equation: $\frac{2y + x}{8} - \frac{3y - x}{4} = 12$.
Multiply through by 8 to clear the denominators:

$8 \left( \frac{2y + x}{8} \right) - 8 \left( \frac{3y - x}{4} \right) = 8 \times 12$

Simplifying each term gives:

$2y + x - 2(3y - x) = 96$

Expanding the terms:

$2y + x - 6y + 2x = 96$

Combine like terms for a simplified second equation:

$3x - 4y = 96$     [Equation (2)]

Now let's use substitution:

From Equation (2), solve for $x$:

$3x = 4y + 96$

$x = \frac{4y + 96}{3}$

Substitute this expression for $x$ in Equation (1):

$14\left( \frac{4y + 96}{3} \right) - 19y = 600$

Multiply through by 3 to clear the fraction:

$14(4y + 96) - 57y = 1800$

$56y + 1344 - 57y = 1800$

Combine like terms:

$-y + 1344 = 1800$

Solve for $y$:

$-y = 1800 - 1344 = 456$

$y = -456$

Substitute $y = -456$ back into the expression for $x$:

$x = \frac{4(-456) + 96}{3}$

$x = \frac{-1824 + 96}{3}$

$x = \frac{-1728}{3}$

$x = -576$

Thus, the solution for the system is:

$x = -576, y = -456$

After verifying against the provided answer choices, the solution is consistent with choice 4.

Therefore, the solution to the problem is $x = -576, y = -456$.

To solve this system of equations, we'll follow these steps:

• Step 1: Analyze the proportions of the coefficients to check for potential parallelism or redundancy.
• Step 2: Attempt elimination to directly identify any inconsistencies.

Now, let's perform the analysis:
Step 1: Notice the proportionality in both equations:
The first equation is $-8x + 5y = 2$ and the second is $-16x + 10y = 5$. Multiply the first equation by 2:
$-16x + 10y = 4$.
This equation is now equivalent in terms of $x$ and $y$ to the second equation, but with a different constant term.

Step 2: Subtract the modified first equation from the second equation:

This result, $0 = 1$, is a contradiction, indicating that the system is inconsistent.

Therefore, the system of equations has no solution.

To solve the given system of equations using substitution, follow these steps:

First, we simplify the given equations. Let's start with the first equation:

Find a common denominator for fractions on the left side. The common denominator of 8 and 2 is 8:

Simplify inside the fraction:

This is our simplified form for the first equation.

Now, simplify the second equation:

Find a common denominator for the fractions, which is 15:

Distribute and simplify:

Multiply through by 15 to clear the fraction:

Now, we have two simplified equations:

From the first equation, solve for $x$:

Substitute $x = \frac{6y + 112}{7}$ into the second equation:

Distribute:

Clear the fraction by multiplying through by 7:

Combine like terms:

Subtract 224 from both sides:

Now, substitute $y \approx 12.93$ back into $x = \frac{6y + 112}{7}$:

Therefore, the solution to the system is:

$x \approx 27.08, \, y \approx 12.93$.

This corresponds to choice 1:

$x = 27.08, y = 12.93$.

To solve the system of equations using the substitution method, follow these detailed steps:

We start with the given system:

Step 1: Clear Fractions

• For the first equation, multiply through by 6 (the least common multiple of 6 and 3):
$(-x + 3y) + 2(5x - y) = 72$ $-x + 3y + 10x - 2y = 72$ $9x + y = 72 \quad \text{(Equation 1)}$
• For the second equation, multiply through by 10 (the LCM of 10 and 5):
$\frac{2x-4y}{10} \times 10 - \frac{-2x-2y}{5} \times 10 = 150$ $(2x - 4y) + (-4x - 4y) = 150$ $2x - 4y + 4x + 4y = 150$ $6x = 150$ $x = 25 \quad \text{(Equation 2)}$

Step 2: Substitute & Solve

Since we have $x = 25$ from Equation 2, substitute $x = 25$ into Equation 1:

Therefore, the solution to the system is $\boxed{x = 25, y = -153}$.