QuestionFind the value of $z$ given $x = z$ , $x = 6k + 13$ , and $y = 8k - 29$ with lines $m$ and $n$ parallel.

Studdy Solution

STEP 1

Assumptions1. Lines $m$ and $n$ are parallel. . A transversal intersects lines $m$ and $n$ .
3. $x$ , $y$ , and $z$ represent the measures of certain angles in the figure.
4. $x = z$ .
5. $x =6k +13$ .
6. $y =8k -29$ .

STEP 2

Since lines $m$ and $n$ are parallel and a transversal intersects them, we can use the property of alternate interior angles to set up an equation. Alternate interior angles are equal when two lines are parallel and intersected by a transversal. Therefore, we can say that $x = y$ .
$x = y$

STEP 3

Now, plug in the given expressions for $x$ and $y$ into the equation.
$6k +13 =8k -29$

STEP 4

To solve for $k$ , we need to get all terms involving $k$ on one side of the equation and the constants on the other side. We can start by subtracting $6k$ from both sides of the equation.
$6k -6k +13 =8k -6k -29$ This simplifies to $13 =2k -29$

STEP 5

Next, add $29$ to both sides of the equation to isolate the term with $k$ .
$13 +29 =2k -29 +29$ This simplifies to $42 =2k$

STEP 6

Finally, divide both sides of the equation by $2$ to solve for $k$ .
$42 /2 =2k /2$ This gives us $k =21$

STEP 7

Now that we have the value of $k$ , we can substitute it back into the equation for $x$ (or $z$ , since $x = z$ ) to find the value of $z$ .
$z =6k +13$

STEP 8

Plug in the value of $k$ into the equation.
$z =6(21) +13$

STEP 9

Calculate the value of $z$ .
$z =6(21) +13 =126 +13 =139$ The value of $z$ is $139$ .

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QuestionFind the value of $z$ given $x = z$ , $x = 6k + 13$ , and $y = 8k - 29$ with lines $m$ and $n$ parallel.

Studdy Solution

STEP 1

Assumptions1. Lines $m$ and $n$ are parallel. . A transversal intersects lines $m$ and $n$ .
3. $x$ , $y$ , and $z$ represent the measures of certain angles in the figure.
4. $x = z$ .
5. $x =6k +13$ .
6. $y =8k -29$ .

STEP 2

Since lines $m$ and $n$ are parallel and a transversal intersects them, we can use the property of alternate interior angles to set up an equation. Alternate interior angles are equal when two lines are parallel and intersected by a transversal. Therefore, we can say that $x = y$ .
$x = y$

STEP 3

Now, plug in the given expressions for $x$ and $y$ into the equation.
$6k +13 =8k -29$

STEP 4

To solve for $k$ , we need to get all terms involving $k$ on one side of the equation and the constants on the other side. We can start by subtracting $6k$ from both sides of the equation.
$6k -6k +13 =8k -6k -29$ This simplifies to $13 =2k -29$

STEP 5

Next, add $29$ to both sides of the equation to isolate the term with $k$ .
$13 +29 =2k -29 +29$ This simplifies to $42 =2k$

STEP 6

Finally, divide both sides of the equation by $2$ to solve for $k$ .
$42 /2 =2k /2$ This gives us $k =21$

STEP 7

Now that we have the value of $k$ , we can substitute it back into the equation for $x$ (or $z$ , since $x = z$ ) to find the value of $z$ .
$z =6k +13$

STEP 8

Plug in the value of $k$ into the equation.
$z =6(21) +13$

STEP 9

Calculate the value of $z$ .
$z =6(21) +13 =126 +13 =139$ The value of $z$ is $139$ .

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QuestionFind the value of zzz given x=zx = zx=z, x=6k+13x = 6k + 13x=6k+13, and y=8k−29y = 8k - 29y=8k−29 with lines mmm and nnn parallel.

Studdy Solution

STEP 1

Assumptions1. Lines mmm and nnn are parallel. . A transversal intersects lines mmm and nnn.3. xxx, yyy, and zzz represent the measures of certain angles in the figure.4. x=zx = zx=z.5. x=6k+13x =6k +13x=6k+13.6. y=8k−29y =8k -29y=8k−29.

STEP 2

Since lines mmm and nnn are parallel and a transversal intersects them, we can use the property of alternate interior angles to set up an equation. Alternate interior angles are equal when two lines are parallel and intersected by a transversal. Therefore, we can say that x=yx = yx=y.x=yx = yx=y

STEP 3

Now, plug in the given expressions for xxx and yyy into the equation.6k+13=8k−296k +13 =8k -296k+13=8k−29

STEP 4

STEP 5

Next, add 292929 to both sides of the equation to isolate the term with kkk.13+29=2k−29+2913 +29 =2k -29 +2913+29=2k−29+29This simplifies to42=2k42 =2k42=2k

STEP 6

Finally, divide both sides of the equation by 222 to solve for kkk.42/2=2k/242 /2 =2k /242/2=2k/2This gives usk=21k =21k=21

STEP 7

Now that we have the value of kkk, we can substitute it back into the equation for xxx (or zzz, since x=zx = zx=z) to find the value of zzz.z=6k+13z =6k +13z=6k+13

STEP 8

Plug in the value of kkk into the equation.z=6(21)+13z =6(21) +13z=6(21)+13

STEP 9

Calculate the value of zzz.z=6(21)+13=126+13=139z =6(21) +13 =126 +13 =139z=6(21)+13=126+13=139The value of zzz is 139139139.

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QuestionFind the value of $z$ given $x = z$ , $x = 6k + 13$ , and $y = 8k - 29$ with lines $m$ and $n$ parallel.

Assumptions1. Lines $m$ and $n$ are parallel. . A transversal intersects lines $m$ and $n$ .
3. $x$ , $y$ , and $z$ represent the measures of certain angles in the figure.
4. $x = z$ .
5. $x =6k +13$ .
6. $y =8k -29$ .

Since lines $m$ and $n$ are parallel and a transversal intersects them, we can use the property of alternate interior angles to set up an equation. Alternate interior angles are equal when two lines are parallel and intersected by a transversal. Therefore, we can say that $x = y$ .
$x = y$

Now, plug in the given expressions for $x$ and $y$ into the equation.
$6k +13 =8k -29$

Next, add $29$ to both sides of the equation to isolate the term with $k$ .
$13 +29 =2k -29 +29$ This simplifies to $42 =2k$

Finally, divide both sides of the equation by $2$ to solve for $k$ .
$42 /2 =2k /2$ This gives us $k =21$

Now that we have the value of $k$ , we can substitute it back into the equation for $x$ (or $z$ , since $x = z$ ) to find the value of $z$ .
$z =6k +13$

Plug in the value of $k$ into the equation.
$z =6(21) +13$

Calculate the value of $z$ .
$z =6(21) +13 =126 +13 =139$ The value of $z$ is $139$ .