If 3𝑥 + 5𝑦 = 15 and 𝑥 − 2𝑦 = 10, then 2𝑥 + 7𝑦 =
Note that
2x+7y = (3x+5y) - (x-2y) = 15-10 = 5
To solve the system of equations:
1) Start by solving for one variable in terms of the other using either substitution or elimination method.
Using the elimination method, let's start by multiplying the second equation by 3 to make the coefficients of 𝑥 in both equations match:
3(𝑥 − 2𝑦) = 10
Simplifying this equation, we get:
3𝑥 − 6𝑦 = 10
Now, we have the following two equations:
3𝑥 + 5𝑦 = 15
3𝑥 − 6𝑦 = 10
2) Subtract the second equation from the first equation:
(3𝑥 + 5𝑦) - (3𝑥 − 6𝑦) = 15 - 10
Simplifying this equation, we have:
11𝑦 = 5
3) Solve for 𝑦 by dividing both sides of the equation by 11:
11𝑦/11 = 5/11
Simplifying this equation, we get:
𝑦 = 5/11
4) Substitute the value of 𝑦 in either of the original equations to solve for 𝑥.
Let's take the first equation:
3𝑥 + 5(𝑦) = 15
Substituting the value of 𝑦 as 5/11, we get:
3𝑥 + 5(5/11) = 15
Simplifying this equation, we have:
3𝑥 + 25/11 = 15
5) Solve for 𝑥 by isolating 𝑥:
3𝑥 = 15 - 25/11
Combining the terms on the right side, we get:
3𝑥 = (165 - 25)/11
Simplifying this equation, we have:
3𝑥 = 140/11
6) Divide both sides of the equation by 3 to solve for 𝑥:
(3𝑥)/3 = (140/11)/3
Simplifying this equation, we get:
𝑥 = 140/33
7) Now that we have the values of 𝑥 and 𝑦, substitute them into the given equation: 2𝑥 + 7𝑦.
Substituting 𝑥 as 140/33 and 𝑦 as 5/11, we have:
2(140/33) + 7(5/11)
Simplifying this equation, we get:
280/33 + 35/11
8) Add the fractions by finding a common denominator:
(280/33) + (35/11) = (280/33) + (99/33)
Combining the fraction, we get:
(280 + 99)/33
9) Simplify the numerator:
379/33
Therefore, 2𝑥 + 7𝑦 = 379/33.
To find the value of 2𝑥 + 7𝑦, we need to solve the system of equations:
Equation 1: 3𝑥 + 5𝑦 = 15
Equation 2: 𝑥 − 2𝑦 = 10
We can solve this system using the method of substitution or elimination.
Using the method of substitution, we can solve Equation 2 for 𝑥 and substitute it into Equation 1:
From Equation 2, 𝑥 = 10 + 2𝑦
Substituting this value of 𝑥 into Equation 1:
3(10 + 2𝑦) + 5𝑦 = 15
30 + 6𝑦 + 5𝑦 = 15
11𝑦 = 15 - 30
11𝑦 = -15
𝑦 = -15/11
Now substitute the value of 𝑦 back into Equation 2 to find 𝑥:
𝑥 - 2(-15/11) = 10
𝑥 + 30/11 = 10
𝑥 = 10 - 30/11
𝑥 = 110/11 - 30/11
𝑥 = 80/11
Finally, substitute the values of 𝑥 and 𝑦 into the expression 2𝑥 + 7𝑦:
2(80/11) + 7(-15/11) = 160/11 - 105/11
= 55/11
= 5
Therefore, the value of 2𝑥 + 7𝑦 is 5.