Solved on Jan 07, 2024

Eliminate x-terms in system: $3x+5y=5$ , multiply by ? on both sides; $-2x-8y=-6$ , multiply by ? on both sides.

STEP 1

Assumptions
1. We have two linear equations: - $3x + 5y = 5$ - $-2x - 8y = -6$
2. We want to eliminate the $x$ -terms from the system of equations.
3. To eliminate the $x$ -terms, we need to find multipliers for each equation such that the coefficients of $x$ in both equations become additive inverses of each other.

STEP 2

Let's denote the multipliers for the first and second equations as $m_1$ and $m_2$ , respectively.

STEP 3

The goal is to make the coefficients of $x$ in both equations equal in magnitude but opposite in sign. This means we need to find $m_1$ and $m_2$ such that:
$m_1 \cdot 3x = -m_2 \cdot (-2x)$

STEP 4

Solve the equation for $m_1$ and $m_2$ :
$m_1 \cdot 3 = m_2 \cdot 2$

STEP 5

We can choose the smallest positive integers for $m_1$ and $m_2$ that satisfy the equation. To do this, we can set $m_1$ equal to 2 and $m_2$ equal to 3, as these are the smallest integers that make the equation true:
$m_1 = 2, \quad m_2 = 3$

STEP 6

Now, multiply the first equation by $m_1$ and the second equation by $m_2$ :
First equation: $3x + 5y = 5$ multiplied by 2 becomes $6x + 10y = 10$ .
Second equation: $-2x - 8y = -6$ multiplied by 3 becomes $-6x - 24y = -18$ .

STEP 7

The new system of equations after applying the multipliers is:
$6x + 10y = 10$ $-6x - 24y = -18$

STEP 8

Adding these two equations together will eliminate the $x$ -terms:
$(6x + 10y) + (-6x - 24y) = 10 + (-18)$

STEP 9

Perform the addition:
$6x - 6x + 10y - 24y = 10 - 18$

STEP 10

Simplify the equation:
$0x - 14y = -8$

STEP 11

The $x$ -terms have been successfully eliminated, leaving us with the equation:
$-14y = -8$
To summarize, to eliminate the $x$ -terms in the given system, we should multiply the first equation by 2 and the second equation by 3.

Was this helpful?

Eliminate x-terms in system: $3x+5y=5$ , multiply by ? on both sides; $-2x-8y=-6$ , multiply by ? on both sides.

STEP 1

STEP 2

Let's denote the multipliers for the first and second equations as $m_1$ and $m_2$ , respectively.

STEP 3

The goal is to make the coefficients of $x$ in both equations equal in magnitude but opposite in sign. This means we need to find $m_1$ and $m_2$ such that:
$m_1 \cdot 3x = -m_2 \cdot (-2x)$

STEP 4

Solve the equation for $m_1$ and $m_2$ :
$m_1 \cdot 3 = m_2 \cdot 2$

STEP 5

We can choose the smallest positive integers for $m_1$ and $m_2$ that satisfy the equation. To do this, we can set $m_1$ equal to 2 and $m_2$ equal to 3, as these are the smallest integers that make the equation true:
$m_1 = 2, \quad m_2 = 3$

STEP 6

Now, multiply the first equation by $m_1$ and the second equation by $m_2$ :
First equation: $3x + 5y = 5$ multiplied by 2 becomes $6x + 10y = 10$ .
Second equation: $-2x - 8y = -6$ multiplied by 3 becomes $-6x - 24y = -18$ .

STEP 7

The new system of equations after applying the multipliers is:
$6x + 10y = 10$ $-6x - 24y = -18$

STEP 8

Adding these two equations together will eliminate the $x$ -terms:
$(6x + 10y) + (-6x - 24y) = 10 + (-18)$

STEP 9

Perform the addition:
$6x - 6x + 10y - 24y = 10 - 18$

STEP 10

Simplify the equation:
$0x - 14y = -8$

STEP 11

The $x$ -terms have been successfully eliminated, leaving us with the equation:
$-14y = -8$
To summarize, to eliminate the $x$ -terms in the given system, we should multiply the first equation by 2 and the second equation by 3.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Eliminate x-terms in system: 3x+5y=53x+5y=53x+5y=5, multiply by ? on both sides; −2x−8y=−6-2x-8y=-6−2x−8y=−6, multiply by ? on both sides.

STEP 1

STEP 2

Let's denote the multipliers for the first and second equations as m1m_1m1​ and m2m_2m2​, respectively.

STEP 3

The goal is to make the coefficients of xxx in both equations equal in magnitude but opposite in sign. This means we need to find m1m_1m1​ and m2m_2m2​ such that:m1⋅3x=−m2⋅(−2x)m_1 \cdot 3x = -m_2 \cdot (-2x)m1​⋅3x=−m2​⋅(−2x)

STEP 4

Solve the equation for m1m_1m1​ and m2m_2m2​:m1⋅3=m2⋅2m_1 \cdot 3 = m_2 \cdot 2m1​⋅3=m2​⋅2

STEP 5

We can choose the smallest positive integers for m1m_1m1​ and m2m_2m2​ that satisfy the equation. To do this, we can set m1m_1m1​ equal to 2 and m2m_2m2​ equal to 3, as these are the smallest integers that make the equation true:m1=2,m2=3m_1 = 2, \quad m_2 = 3m1​=2,m2​=3

STEP 6

STEP 7

The new system of equations after applying the multipliers is:6x+10y=106x + 10y = 106x+10y=10 −6x−24y=−18-6x - 24y = -18−6x−24y=−18

STEP 8

Adding these two equations together will eliminate the xxx-terms:(6x+10y)+(−6x−24y)=10+(−18)(6x + 10y) + (-6x - 24y) = 10 + (-18)(6x+10y)+(−6x−24y)=10+(−18)

STEP 9

Perform the addition:6x−6x+10y−24y=10−186x - 6x + 10y - 24y = 10 - 186x−6x+10y−24y=10−18

STEP 10

Simplify the equation:0x−14y=−80x - 14y = -80x−14y=−8

STEP 11

The xxx-terms have been successfully eliminated, leaving us with the equation:−14y=−8-14y = -8−14y=−8To summarize, to eliminate the xxx-terms in the given system, we should multiply the first equation by 2 and the second equation by 3.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Eliminate x-terms in system: $3x+5y=5$ , multiply by ? on both sides; $-2x-8y=-6$ , multiply by ? on both sides.

Let's denote the multipliers for the first and second equations as $m_1$ and $m_2$ , respectively.

The goal is to make the coefficients of $x$ in both equations equal in magnitude but opposite in sign. This means we need to find $m_1$ and $m_2$ such that:
$m_1 \cdot 3x = -m_2 \cdot (-2x)$

Solve the equation for $m_1$ and $m_2$ :
$m_1 \cdot 3 = m_2 \cdot 2$

We can choose the smallest positive integers for $m_1$ and $m_2$ that satisfy the equation. To do this, we can set $m_1$ equal to 2 and $m_2$ equal to 3, as these are the smallest integers that make the equation true:
$m_1 = 2, \quad m_2 = 3$

The new system of equations after applying the multipliers is:
$6x + 10y = 10$ $-6x - 24y = -18$

Adding these two equations together will eliminate the $x$ -terms:
$(6x + 10y) + (-6x - 24y) = 10 + (-18)$

Perform the addition:
$6x - 6x + 10y - 24y = 10 - 18$

Simplify the equation:
$0x - 14y = -8$

The $x$ -terms have been successfully eliminated, leaving us with the equation:
$-14y = -8$
To summarize, to eliminate the $x$ -terms in the given system, we should multiply the first equation by 2 and the second equation by 3.