Question

11 practice problems
diego has $11 and begins saving $5 each week toward buying a new phone. at the same time that diego begins saving, lin has $60 and begins spending $2 per week on supplies for her art class. is there a week when they have the same amount of money? how much do they have at that time?
use a graph to find $x$ and $y$ values that make both $y = -\frac{2}{3}x + 3$ and $y = 2x - 5$ true.
ere the graphs of two equations intersect has $y$-coordina
find the other equation if its graph has a slope of 1.

Explanation:

First Problem:

Step1: Define variables & equations

Let $x$ = number of weeks, $y$ = total money.
Diego: $y = 5x + 11$
Lin: $y = -2x + 60$

Step2: Set equations equal

$5x + 11 = -2x + 60$

Step3: Solve for $x$

$5x + 2x = 60 - 11$
$7x = 49$
$x = 7$

Step4: Find $y$ value

$y = 5(7) + 11 = 35 + 11 = 46$

Step1: Set equations equal

$-\frac{2}{3}x + 3 = 2x - 5$

Step2: Eliminate fraction, solve $x$

Multiply all terms by 3:
$-2x + 9 = 6x - 15$
$-2x -6x = -15 -9$
$-8x = -24$
$x = 3$

Step3: Find $y$ value

$y = 2(3) - 5 = 6 - 5 = 1$

Step1: Use point-slope form

Intersection point is $(3,1)$, slope $m=1$.
Point-slope formula: $y - y_1 = m(x - x_1)$

Step2: Substitute values

$y - 1 = 1(x - 3)$

Step3: Simplify to slope-intercept

$y = x - 3 + 1$
$y = x - 2$

Answer:

Yes, they have the same amount of money after 7 weeks, and each has \$46.

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Second Problem: