Question

1. jemila is thinking of a number such that the sum of the number and five times two more than the number is 26 more than four times the number. determine the number jemila is thinking of.
2. as if #2 wasnt confusing enough, jemila is now trying to come up with a number where three less than 8 times the number is equal to half of 16 times the number after it was increased by 1. she cant seem to find a number that works. explain why.

when finding the intersection of two lines from both algebra i and geometry, you first \set the linear equations equal\ to each other. find the intersection point of the two lines whose equations are shown below. be sure to find both the x and y coordinates.
y = 5x + 1 and y = 2x - 11

1. explain why you cannot find the intersection points of the two lines shown below. try to give both an algebraic reason based on #4 and a graphical reason (hint: the two lines have the same slope).

y = 4x + 1 and y = 4x + 10

Explanation:

2.

Step1: Translate to an equation

Let the number be $x$. The sum of the number and five times two more than the number is $x + 5(x + 2)$. And it is equal to 26 more than four times the number, so the equation is $x+5(x + 2)=4x + 26$.

Step2: Expand the left - hand side

$x+5x+10 = 4x + 26$.
Combining like terms on the left - hand side gives $6x+10=4x + 26$.

Step3: Isolate the variable terms

Subtract $4x$ from both sides: $6x-4x+10=4x-4x + 26$, which simplifies to $2x+10 = 26$.

Step4: Solve for $x$

Subtract 10 from both sides: $2x+10 - 10=26 - 10$, so $2x=16$. Then divide both sides by 2: $x = 8$.

Step1: Translate to an equation

Let the number be $x$. Three less than 8 times the number is $8x-3$. Half of 16 times the number after it was increased by 1 is $\frac{1}{2}\times16(x + 1)$. The equation is $8x-3=\frac{1}{2}\times16(x + 1)$.

Step2: Simplify the right - hand side

$\frac{1}{2}\times16(x + 1)=8(x + 1)=8x+8$.
So the equation becomes $8x-3=8x + 8$.

Step3: Analyze the equation

Subtract $8x$ from both sides: $8x-8x-3=8x-8x + 8$, which gives $-3=8$. This is a contradiction. So there is no solution because the resulting equation is false for all values of $x$.

Step1: Set the equations equal

Since $y = 5x+1$ and $y = 2x-11$, we set $5x+1=2x-11$.

Step2: Solve for $x$

Subtract $2x$ from both sides: $5x-2x+1=2x-2x-11$, so $3x+1=-11$.
Subtract 1 from both sides: $3x+1 - 1=-11 - 1$, so $3x=-12$.
Divide both sides by 3: $x=-4$.

Step3: Solve for $y$

Substitute $x = - 4$ into $y = 5x+1$. Then $y=5\times(-4)+1=-20 + 1=-19$.

Answer:

8

3.