Proportionality>Writing Linear Equations in the Form y = mx + b from Multiple Representations(TEKS.Math.8.5.I)
Help Questions
Texas 8th Grade Math › Proportionality>Writing Linear Equations in the Form y = mx + b from Multiple Representations(TEKS.Math.8.5.I)
A gym membership costs 50 to join plus 30 per month. Let $x$ be the number of months and let $y$ be the total cost in dollars.
Which equation in $y=mx+b$ form models this relationship?
$y=50x+30$
$x=30y+50$
$y=30x+50$
$y=30+50x$
Explanation
The rate of change (slope) is the monthly fee, 30 dollars per month, so $m=30$. The starting value (y-intercept) is the one-time joining fee, 50 dollars, so $b=50$. Therefore $y=30x+50$. This connects to the context: $30x$ is the total monthly charges and $50$ is the initial fee.
On a coordinate plane, a line crosses the $y$-axis at $-2$ and passes through the point $(3,4)$.
Which equation in $y=mx+b$ form models this line?
$y=-2x+2$
$y=\tfrac{1}{2}x-2$
$x=2y-2$
$y=2x-2$
Explanation
The y-intercept is $b=-2$. Using $(0,-2)$ and $(3,4)$, the slope is $m=\dfrac{4-(-2)}{3-0}=\dfrac{6}{3}=2$. So the equation is $y=2x-2$. Here $m$ is the rise over run from the graph and $b$ is where the line crosses the $y$-axis.
A linear relationship between two quantities is described by these values: when $x=2$, $y=11$; when $x=5$, $y=20$.
Write an equation in $y=mx+b$ form for this relationship.
$y=2x+7$
$y=3x+5$
$x=3y+5$
$y=3x-5$
Explanation
Find the slope: $m=\dfrac{20-11}{5-2}=\dfrac{9}{3}=3$. Substitute into $y=mx+b$ using $(2,11)$: $11=3(2)+b\Rightarrow 11=6+b\Rightarrow b=5$. So $y=3x+5$. The slope is the rate of change between the pairs, and the intercept is the starting value when $x=0$.