Solve the system of two linear equations.

Question

Solve the system of two linear equations.

- 3x - 4y = - 23
3x - 5y = 5 (1 point)

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3 answers
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Answer 1

Bot GPT 3.5 answered
1 year ago

To solve the system of two linear equations, we'll use the method of elimination.

First, let's multiply the second equation by -1 to make the coefficients of x in both equations opposites:

-1(3x - 5y) = -1(5)
-3x + 5y = -5

Now we have these two equations:

-3x - 4y = -23
-3x + 5y = -5

Next, we'll add these two equations together to eliminate x:

(-3x - 4y) + (-3x + 5y) = -23 + (-5)
-3x - 3x - 4y + 5y = -23 - 5
-6x + y = -28

Now, we'll isolate y in this equation:

y = -6x - 28

We can substitute this expression for y in one of the original equations. Let's use the first equation:

-3x - 4(-6x - 28) = -23
-3x + 24x + 112 = -23
21x + 112 = -23
21x = -23 - 112
21x = -135
x = -135/21
x = -45/7

Now, substitute this value for x back into the expression for y:

y = -6(-45/7) - 28
y = 270/7 - 196/7
y = 74/7

Therefore, the solution to the system of equations is x = -45/7 and y = 74/7.

Answer 2

Step-by-Step Bot answered
1 year ago

To solve the system of two linear equations, we can use the method of elimination.

First, let's write down the given equations:

Equation 1: -3x - 4y = -23
Equation 2: 3x - 5y = 5

To eliminate one of the variables, we'll multiply both sides of Equation 1 by 3 and Equation 2 by -3. This will make the x coefficients in both equations opposite and will allow us to eliminate x when we add the equations together.

Multiplying Equation 1 by 3:
3 * (-3x - 4y) = 3 * (-23)
-9x - 12y = -69

Multiplying Equation 2 by -3:
-3 * (3x - 5y) = -3 * 5
-9x + 15y = -15

Now, we can add the two equations together to eliminate x:
(-9x - 12y) + (-9x + 15y) = -69 + (-15)
-9x - 9x - 12y + 15y = -69 - 15
-18x + 3y = -84

Now, we have a new equation in terms of y:

-18x + 3y = -84

To find the value of y, we isolate y by rearranging the equation:

3y = -18x - 84
y = (-18x - 84) / 3
y = -6x - 28

Now, we substitute this value of y back into one of the original equations, let's use Equation 1:

-3x - 4(-6x - 28) = -23
-3x + 24x + 112 = -23
21x + 112 = -23
21x = -23 - 112
21x = -135
x = -135/21
x = -45/7

So, the solution to the system of equations is:
x = -45/7
y = -6x - 28 or y = -6(-45/7) - 28

Therefore, the solution to the system of equations is:
x = -45/7
y = 62/7

Answer 3

Explain Bot answered
1 year ago

To solve the system of linear equations, you can use the method of elimination or substitution. I will explain both methods, and you can choose the one you prefer.

1) Method of Elimination:
Step 1: Multiply one or both of the equations by appropriate constants to make the coefficients of either x or y the same or multiples of each other. In this case, it is easier to eliminate the x-term by multiplying the first equation by 3 and the second equation by -3:
-3x - 4y = -23 *3
3x - 5y = 5 *-3
Gives:
-9x - 12y = -69 (equation A)
-9x + 15y = -15 (equation B)

Step 2: Add equation A and B together to eliminate the x-term. This will give you a new equation with only y:
(-9x - 12y) + (-9x + 15y) = -69 + (-15)
-18y = -84

Step 3: Solve the new equation for y:
Divide both sides of the equation by -18:
y = -84 / -18
y = 14/3 or approximately 4.67

Step 4: Now, substitute the value of y into either of the original two equations. Let's use the first equation:
-3x - 4(14/3) = -23

Step 5: Solve for x:
-3x - 56/3 = -23
-3x = -23 + 56/3
-3x = -69/3 + 56/3
-3x = -13/3

Divide both sides of the equation by -3:
x = (-13/3) / -3
x = 13/9 or approximately 1.44

Thus, the solution to the system of equations is x = 13/9 and y = 14/3.

2) Method of Substitution:
Step 1: Solve one of the equations for one variable in terms of the other variable. Let's solve the second equation for x in terms of y:
3x - 5y = 5
3x = 5 + 5y

Divide both sides of the equation by 3:
x = (5 + 5y) / 3

Step 2: Substitute the expression for x into the other equation:
-3((5 + 5y) / 3) - 4y = -23

Step 3: Simplify and solve for y:
-5 - 5y - 4y = -23
-5y - 4y = -23 + 5
-9y = -18

Divide both sides of the equation by -9:
y = -18 / -9
y = 2

Step 4: Substitute the value of y back into one of the original equations to solve for x. Let's use the first equation:
-3x - 4(2) = -23

Simplify and solve for x:
-3x - 8 = -23
-3x = -23 + 8
-3x = -15

Divide both sides of the equation by -3:
x = -15 / -3
x = 5

Therefore, the solution to the system of equations is x = 5 and y = 2.