Solve the system of two linear equations.
−3x−4y=−23
3x−5y=5 (1 point)
Responses
(5, 2)
left parenthesis 5 comma 2 right parenthesis
(5,−2)
left parenthesis 5 comma negative 2 right parenthesis
(−5,−2)
left parenthesis negative 5 comma negative 2 right parenthesis
(−5, 2)
To solve the system of two linear equations, we can use the method of elimination. Adding the two equations together eliminates the x-term:
(-3x - 4y) + (3x - 5y) = -23 + 5
-3x + 3x - 4y - 5y = -18
-9y = -18
Dividing both sides by -9:
y = 2
Substituting this value of y into one of the original equations, we can solve for x:
-3x - 4(2) = -23
-3x - 8 = -23
-3x = -15
x = 5
Therefore, the solution to the system of equations is (5, 2).
To solve the system of equations −3x−4y=−23 and 3x−5y=5:
Step 1: Multiply the first equation by 3 and the second equation by -3 to eliminate x:
-9x - 12y = -69
-9x + 15y = -15
Step 2: Add the two resulting equations together to eliminate x:
(-9x - 12y) + (-9x + 15y) = -69 + (-15)
-18y = -84
Step 3: Divide both sides of the equation by -18 to solve for y:
y = -84 / -18
y = 4.67 (rounded to two decimal places)
Step 4: Substitute the value of y back into one of the original equations, for example, the second equation:
3x - 5 * 4.67 = 5
Step 5: Simplify and solve for x:
3x - 23.35 = 5
3x = 28.35
x = 9.45 (rounded to two decimal places)
So the solution to the system of equations is (9.45, 4.67).
To solve this system of equations, we can use the elimination or substitution method. Let's use the elimination method:
First, let's rearrange the second equation to make the x-coefficient the same as the first equation:
3x - 5y = 5
Multiply both sides of the second equation by -1:
-3x + 5y = -5
Now, we can add the two equations:
(-3x - 4y) + (-3x + 5y) = (-23) + (-5)
Simplifying:
-6x + y = -28
Now we have a new equation. Let's call it equation (3).
Next, we'll solve equations (1) and (3) as a system of equations. We'll multiply both sides of equation (1) by 6 to make the x-coefficient in both equations the same:
-18x - 24y = -138
Now we have two equations:
-18x - 24y = -138 (equation 4)
-6x + y = -28 (equation 3)
To eliminate the y variable, we'll multiply equation (3) by 24, and equation (4) by -1:
-24(-6x + y) = -24(-28) becomes 144x - 24y = 672 (equation 5)
x is eliminated in equation (4):
18x + 24y = 138 (equation 6)
Now, let's add equations (5) and (6):
(144x - 24y) + (18x + 24y) = 672 + 138
Simplifying:
162x = 810
Divide both sides by 162:
x = 5
Now, substitute the value of x into equation (3):
-6(5) + y = -28
Simplifying:
-30 + y = -28
Add 30 to both sides:
y = -28 + 30
Simplifying:
y = 2
Therefore, the solution to the system of equations is (x, y) = (5, 2).