Solved on Jan 20, 2024

Solve the system of linear equations: $\frac{|4 x-2|}{5}=10$ and $-2|x-7|=-12$ .

STEP 1

Assumptions
1. We are given two separate equations to solve for $x$ .
2. The absolute value function $| \cdot |$ is defined such that $|a| = a$ if $a \geq 0$ and $|a| = -a$ if $a < 0$ .
3. We will solve each equation separately and then check for common solutions if any.

STEP 2

Let's start by solving the first equation:
$\frac{|4x - 2|}{5} = 10$

STEP 3

Multiply both sides of the equation by 5 to eliminate the denominator.
$|4x - 2| = 5 \times 10$

STEP 4

Calculate the product on the right side of the equation.
$|4x - 2| = 50$

STEP 5

The absolute value equation $|4x - 2| = 50$ can be split into two separate equations:
1. $4x - 2 = 50$
2. $4x - 2 = -50$
We will solve each equation separately.

STEP 6

Solve the first equation $4x - 2 = 50$ .
Add 2 to both sides of the equation.
$4x = 50 + 2$

STEP 7

Calculate the sum on the right side of the equation.
$4x = 52$

STEP 8

Divide both sides of the equation by 4 to solve for $x$ .
$x = \frac{52}{4}$

STEP 9

Calculate the quotient to find the value of $x$ .
$x = 13$

STEP 10

Now, solve the second equation $4x - 2 = -50$ .
Add 2 to both sides of the equation.
$4x = -50 + 2$

STEP 11

Calculate the sum on the right side of the equation.
$4x = -48$

STEP 12

Divide both sides of the equation by 4 to solve for $x$ .
$x = \frac{-48}{4}$

STEP 13

Calculate the quotient to find the value of $x$ .
$x = -12$

STEP 14

The solutions for the first equation are $x = 13$ and $x = -12$ . Now let's solve the second equation:
$-2|x - 7| = -12$

STEP 15

Divide both sides of the equation by -2 to isolate the absolute value expression.
$|x - 7| = \frac{-12}{-2}$

STEP 16

Calculate the quotient on the right side of the equation.
$|x - 7| = 6$

STEP 17

The absolute value equation $|x - 7| = 6$ can be split into two separate equations:
1. $x - 7 = 6$
2. $x - 7 = -6$
We will solve each equation separately.

STEP 18

Solve the first equation $x - 7 = 6$ .
Add 7 to both sides of the equation.
$x = 6 + 7$

STEP 19

Calculate the sum to find the value of $x$ .
$x = 13$

STEP 20

Now, solve the second equation $x - 7 = -6$ .
Add 7 to both sides of the equation.
$x = -6 + 7$

STEP 21

Calculate the sum to find the value of $x$ .
$x = 1$

STEP 22

The solutions for the second equation are $x = 13$ and $x = 1$ .

STEP 23

Comparing the solutions from both equations, we find that $x = 13$ is a common solution.

STEP 24

The final solution to the system of equations is $x = 13$ .
$x = 13$ is the only value that satisfies both equations.

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Solve the system of linear equations: ∣4x−2∣5=10\frac{|4 x-2|}{5}=105∣4x−2∣​=10 and −2∣x−7∣=−12-2|x-7|=-12−2∣x−7∣=−12.

STEP 1

STEP 2

Let's start by solving the first equation:∣4x−2∣5=10\frac{|4x - 2|}{5} = 105∣4x−2∣​=10

STEP 3

Multiply both sides of the equation by 5 to eliminate the denominator.∣4x−2∣=5×10|4x - 2| = 5 \times 10∣4x−2∣=5×10

STEP 4

Calculate the product on the right side of the equation.∣4x−2∣=50|4x - 2| = 50∣4x−2∣=50

STEP 5

The absolute value equation ∣4x−2∣=50|4x - 2| = 50∣4x−2∣=50 can be split into two separate equations:1. 4x−2=504x - 2 = 504x−2=502. 4x−2=−504x - 2 = -504x−2=−50We will solve each equation separately.

STEP 6

Solve the first equation 4x−2=504x - 2 = 504x−2=50.Add 2 to both sides of the equation.4x=50+24x = 50 + 24x=50+2

STEP 7

Calculate the sum on the right side of the equation.4x=524x = 524x=52

STEP 8

Divide both sides of the equation by 4 to solve for xxx.x=524x = \frac{52}{4}x=452​

STEP 9

Calculate the quotient to find the value of xxx.x=13x = 13x=13

STEP 10

Now, solve the second equation 4x−2=−504x - 2 = -504x−2=−50.Add 2 to both sides of the equation.4x=−50+24x = -50 + 24x=−50+2

STEP 11

Calculate the sum on the right side of the equation.4x=−484x = -484x=−48

STEP 12

Divide both sides of the equation by 4 to solve for xxx.x=−484x = \frac{-48}{4}x=4−48​

STEP 13

Calculate the quotient to find the value of xxx.x=−12x = -12x=−12

STEP 14

The solutions for the first equation are x=13x = 13x=13 and x=−12x = -12x=−12. Now let's solve the second equation:−2∣x−7∣=−12-2|x - 7| = -12−2∣x−7∣=−12

STEP 15

Divide both sides of the equation by -2 to isolate the absolute value expression.∣x−7∣=−12−2|x - 7| = \frac{-12}{-2}∣x−7∣=−2−12​

STEP 16

Calculate the quotient on the right side of the equation.∣x−7∣=6|x - 7| = 6∣x−7∣=6

STEP 17

The absolute value equation ∣x−7∣=6|x - 7| = 6∣x−7∣=6 can be split into two separate equations:1. x−7=6x - 7 = 6x−7=62. x−7=−6x - 7 = -6x−7=−6We will solve each equation separately.

STEP 18

Solve the first equation x−7=6x - 7 = 6x−7=6.Add 7 to both sides of the equation.x=6+7x = 6 + 7x=6+7

STEP 19

Calculate the sum to find the value of xxx.x=13x = 13x=13

STEP 20

Now, solve the second equation x−7=−6x - 7 = -6x−7=−6.Add 7 to both sides of the equation.x=−6+7x = -6 + 7x=−6+7

STEP 21

Calculate the sum to find the value of xxx.x=1x = 1x=1

STEP 22

The solutions for the second equation are x=13x = 13x=13 and x=1x = 1x=1.

STEP 23

Comparing the solutions from both equations, we find that x=13x = 13x=13 is a common solution.

STEP 24

The final solution to the system of equations is x=13x = 13x=13.x=13x = 13x=13 is the only value that satisfies both equations.

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Solve the system of linear equations: $\frac{|4 x-2|}{5}=10$ and $-2|x-7|=-12$ .

Let's start by solving the first equation:
$\frac{|4x - 2|}{5} = 10$

Multiply both sides of the equation by 5 to eliminate the denominator.
$|4x - 2| = 5 \times 10$

Calculate the product on the right side of the equation.
$|4x - 2| = 50$

The absolute value equation $|4x - 2| = 50$ can be split into two separate equations:
1. $4x - 2 = 50$
2. $4x - 2 = -50$
We will solve each equation separately.

Solve the first equation $4x - 2 = 50$ .
Add 2 to both sides of the equation.
$4x = 50 + 2$

Calculate the sum on the right side of the equation.
$4x = 52$

Divide both sides of the equation by 4 to solve for $x$ .
$x = \frac{52}{4}$

Calculate the quotient to find the value of $x$ .
$x = 13$

Now, solve the second equation $4x - 2 = -50$ .
Add 2 to both sides of the equation.
$4x = -50 + 2$

Calculate the sum on the right side of the equation.
$4x = -48$

Divide both sides of the equation by 4 to solve for $x$ .
$x = \frac{-48}{4}$

Calculate the quotient to find the value of $x$ .
$x = -12$

The solutions for the first equation are $x = 13$ and $x = -12$ . Now let's solve the second equation:
$-2|x - 7| = -12$

Divide both sides of the equation by -2 to isolate the absolute value expression.
$|x - 7| = \frac{-12}{-2}$

Calculate the quotient on the right side of the equation.
$|x - 7| = 6$

The absolute value equation $|x - 7| = 6$ can be split into two separate equations:
1. $x - 7 = 6$
2. $x - 7 = -6$
We will solve each equation separately.

Solve the first equation $x - 7 = 6$ .
Add 7 to both sides of the equation.
$x = 6 + 7$

Calculate the sum to find the value of $x$ .
$x = 13$

Now, solve the second equation $x - 7 = -6$ .
Add 7 to both sides of the equation.
$x = -6 + 7$

Calculate the sum to find the value of $x$ .
$x = 1$

The solutions for the second equation are $x = 13$ and $x = 1$ .

Comparing the solutions from both equations, we find that $x = 13$ is a common solution.

The final solution to the system of equations is $x = 13$ .
$x = 13$ is the only value that satisfies both equations.