Question Solve for $-11 x-4 y$ given $-4 x-10 y=8$ and $-7 x+6 y=2$ .

Studdy Solution

STEP 1

Assumptions
1. We have two linear equations: $-4x - 10y = 8$ $-7x + 6y = 2$
2. We want to find the value of: $-11x - 4y$

STEP 2

To find the value of $-11x - 4y$ , we first need to solve the system of linear equations for $x$ and $y$ .

STEP 3

We can use the method of substitution or elimination to solve the system. In this case, we will use the elimination method to eliminate one of the variables.

STEP 4

We will first multiply the first equation by 6 and the second equation by 10 to make the coefficients of $y$ opposites.
$6(-4x - 10y) = 6 \cdot 8$ $10(-7x + 6y) = 10 \cdot 2$

STEP 5

Perform the multiplication for both equations.
$-24x - 60y = 48$ $-70x + 60y = 20$

STEP 6

Now, we add the two equations together to eliminate $y$ .
$(-24x - 60y) + (-70x + 60y) = 48 + 20$

STEP 7

Simplify the equation by combining like terms.
$-24x - 70x = -94x$ $48 + 20 = 68$
So we have: $-94x = 68$

STEP 8

Solve for $x$ by dividing both sides of the equation by $-94$ .
$x = \frac{68}{-94}$

STEP 9

Simplify the fraction to find the value of $x$ .
$x = \frac{68}{-94} = \frac{-34}{47}$

STEP 10

Now that we have the value of $x$ , we can substitute it back into one of the original equations to solve for $y$ . We will use the first equation for this purpose.
$-4x - 10y = 8$

STEP 11

Substitute $x = \frac{-34}{47}$ into the equation.
$-4\left(\frac{-34}{47}\right) - 10y = 8$

STEP 12

Multiply $-4$ by $\frac{-34}{47}$ .
$\frac{136}{47} - 10y = 8$

STEP 13

Now, we need to solve for $y$ . First, subtract $\frac{136}{47}$ from both sides of the equation.
$-10y = 8 - \frac{136}{47}$

STEP 14

To subtract the fraction from the whole number, we need to express the whole number as a fraction with the same denominator.
$8 = \frac{8 \cdot 47}{47} = \frac{376}{47}$

STEP 15

Now we can subtract the fractions.
$-10y = \frac{376}{47} - \frac{136}{47}$

STEP 16

Subtract the numerators and keep the common denominator.
$-10y = \frac{376 - 136}{47}$

STEP 17

Calculate the subtraction in the numerator.
$-10y = \frac{240}{47}$

STEP 18

Divide both sides by $-10$ to solve for $y$ .
$y = \frac{240}{47} \cdot \frac{-1}{10}$

STEP 19

Simplify the fraction to find the value of $y$ .
$y = \frac{240}{-470} = \frac{-24}{47}$

STEP 20

Now we have both $x$ and $y$ :
$x = \frac{-34}{47}$ $y = \frac{-24}{47}$

STEP 21

Finally, we can find the value of $-11x - 4y$ by substituting the values of $x$ and $y$ we found.
$-11x - 4y = -11\left(\frac{-34}{47}\right) - 4\left(\frac{-24}{47}\right)$

STEP 22

Multiply $-11$ by $\frac{-34}{47}$ and $-4$ by $\frac{-24}{47}$ .
$-11x - 4y = \frac{374}{47} + \frac{96}{47}$

STEP 23

Add the fractions by adding the numerators and keeping the common denominator.
$-11x - 4y = \frac{374 + 96}{47}$

STEP 24

Calculate the addition in the numerator.
$-11x - 4y = \frac{470}{47}$

STEP 25

Simplify the fraction.
$-11x - 4y = 10$
The value of $-11x - 4y$ is 10.

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