Math / Algebra

Question1. The tables of ordered pairs represent some points on the graphs of two different lines
Line A: \begin{tabular}{|c|c|c|c|c|c|c|} \hline $x$ & 1 & 3 & 5 & 7 & 9 & 11 \\ \hline $y$ & 6 & 18 & 30 & 42 & 54 & 66 \\ \hline \end{tabular}
Line B: \begin{tabular}{|c|c|c|c|c|c|} \hline x & 2 & 4 & 6 & 8 & 10 \\ \hline $\mathbf{y}$ & 4 & 7 & 10 & 13 & 16 \\ \hline \end{tabular}
Which system of equations represents the graph of these two lines? a. $\begin{array}{l} -6 x+y=0 \\ -1.5 x+y=1 \end{array}$ b. $\begin{array}{l} y=0 x+6 \\ -1.5 x+y=1 \end{array}$ c. $\begin{array}{l} -6 x+y=0 \\ y=x+1.5 \end{array}$ d. $\begin{array}{l} y=0 x+6 \\ y=x+1.5 \end{array}$

Studdy Solution

STEP 1

What is this asking? Which pair of equations matches the data in the tables for Line A and Line B? Watch out! Don't mix up the lines!
Make sure each equation matches the right table.

STEP 2

1. Find Equation for Line A
2. Find Equation for Line B
3. Match with Answer Choices

STEP 3

Let's peep at Line A's table!
Notice how when $x$ increases by 2, $y$ increases by 12.
That means the slope is $\frac{12}{2} = 6$ .
So, we're looking for an equation that reflects this rate of change!

STEP 4

Let's test this with the first two points.
We have $(1, 6)$ and $(3, 18)$ .
The slope is $\frac{18 - 6}{3 - 1} = \frac{12}{2} = 6$ .
Awesome!

STEP 5

We can use the point-slope form: $y - y_1 = m(x - x_1)$ .
Let's use the point $(1, 6)$ and our slope $m = 6$ .
So, $y - 6 = 6(x - 1)$ .

STEP 6

Simplify! $y - 6 = 6x - 6$ .
Add 6 to both sides to isolate $y$ : $y = 6x$ .

STEP 7

Now, let's investigate Line B!
When $x$ increases by 2, $y$ increases by 3.
The slope is $\frac{3}{2} = 1.5$ .

STEP 8

Let's verify this with $(2, 4)$ and $(4, 7)$ .
The slope is $\frac{7 - 4}{4 - 2} = \frac{3}{2} = 1.5$ .
Perfect!

STEP 9

Again, we use the point-slope form: $y - y_1 = m(x - x_1)$ .
Let's use the point $(2, 4)$ and our slope $m = 1.5$ .
So, $y - 4 = 1.5(x - 2)$ .

STEP 10

Simplify! $y - 4 = 1.5x - 3$ .
Add 4 to both sides to isolate $y$ : $y = 1.5x + 1$ .

STEP 11

Our equations are $y = 6x$ and $y = 1.5x + 1$ .

STEP 12

Let's rewrite our first equation.
Subtracting $6x$ from both sides gives us $-6x + y = 0$ .

STEP 13

Let's rewrite our second equation.
Subtracting $1.5x$ from both sides gives us $-1.5x + y = 1$ .

Studdy Solution

STEP 1

What is this asking? Which pair of equations matches the data in the tables for Line A and Line B? Watch out! Don't mix up the lines!
Make sure each equation matches the right table.

STEP 2

1. Find Equation for Line A
2. Find Equation for Line B
3. Match with Answer Choices

STEP 3

Let's peep at Line A's table!
Notice how when $x$ increases by 2, $y$ increases by 12.
That means the slope is $\frac{12}{2} = 6$ .
So, we're looking for an equation that reflects this rate of change!

STEP 4

Let's test this with the first two points.
We have $(1, 6)$ and $(3, 18)$ .
The slope is $\frac{18 - 6}{3 - 1} = \frac{12}{2} = 6$ .
Awesome!

STEP 5

We can use the point-slope form: $y - y_1 = m(x - x_1)$ .
Let's use the point $(1, 6)$ and our slope $m = 6$ .
So, $y - 6 = 6(x - 1)$ .

STEP 6

Simplify! $y - 6 = 6x - 6$ .
Add 6 to both sides to isolate $y$ : $y = 6x$ .

STEP 7

Now, let's investigate Line B!
When $x$ increases by 2, $y$ increases by 3.
The slope is $\frac{3}{2} = 1.5$ .

STEP 8

Let's verify this with $(2, 4)$ and $(4, 7)$ .
The slope is $\frac{7 - 4}{4 - 2} = \frac{3}{2} = 1.5$ .
Perfect!

STEP 9

Again, we use the point-slope form: $y - y_1 = m(x - x_1)$ .
Let's use the point $(2, 4)$ and our slope $m = 1.5$ .
So, $y - 4 = 1.5(x - 2)$ .

STEP 10

Simplify! $y - 4 = 1.5x - 3$ .
Add 4 to both sides to isolate $y$ : $y = 1.5x + 1$ .

STEP 11

Our equations are $y = 6x$ and $y = 1.5x + 1$ .

STEP 12

Let's rewrite our first equation.
Subtracting $6x$ from both sides gives us $-6x + y = 0$ .

STEP 13

Let's rewrite our second equation.
Subtracting $1.5x$ from both sides gives us $-1.5x + y = 1$ .

STEP 14

Looking at the answer choices, option (a) matches our equations perfectly!

STEP 15

The correct answer is (a).

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Studdy Solution

STEP 1

What is this asking? Which pair of equations matches the data in the tables for Line A and Line B? Watch out! Don't mix up the lines!Make sure each equation matches the *right* table.

STEP 2

1. Find Equation for Line A2. Find Equation for Line B3. Match with Answer Choices

STEP 3

Let's peep at Line A's table!Notice how when xxx increases by **2**, yyy increases by **12**.That means the *slope* is 122=6\frac{12}{2} = 6212​=6.So, we're looking for an equation that reflects this **rate of change**!

STEP 4

Let's **test** this with the first two points.We have (1,6)(1, 6)(1,6) and (3,18)(3, 18)(3,18).The slope is 18−63−1=122=6\frac{18 - 6}{3 - 1} = \frac{12}{2} = 63−118−6​=212​=6.Awesome!

STEP 5

We can use the *point-slope form*: y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1​=m(x−x1​).Let's use the point (1,6)(1, 6)(1,6) and our slope m=6m = 6m=6.So, y−6=6(x−1)y - 6 = 6(x - 1)y−6=6(x−1).

STEP 6

**Simplify**! y−6=6x−6y - 6 = 6x - 6y−6=6x−6.Add 6 to both sides to *isolate* yyy: y=6xy = 6xy=6x.

STEP 7

Now, let's **investigate** Line B!When xxx increases by **2**, yyy increases by **3**.The *slope* is 32=1.5\frac{3}{2} = 1.523​=1.5.

STEP 8

Let's **verify** this with (2,4)(2, 4)(2,4) and (4,7)(4, 7)(4,7).The slope is 7−44−2=32=1.5\frac{7 - 4}{4 - 2} = \frac{3}{2} = 1.54−27−4​=23​=1.5.Perfect!

STEP 9

Again, we use the *point-slope form*: y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1​=m(x−x1​).Let's use the point (2,4)(2, 4)(2,4) and our slope m=1.5m = 1.5m=1.5.So, y−4=1.5(x−2)y - 4 = 1.5(x - 2)y−4=1.5(x−2).

STEP 10

**Simplify**! y−4=1.5x−3y - 4 = 1.5x - 3y−4=1.5x−3.Add 4 to both sides to *isolate* yyy: y=1.5x+1y = 1.5x + 1y=1.5x+1.

STEP 11

Our equations are y=6xy = 6xy=6x and y=1.5x+1y = 1.5x + 1y=1.5x+1.

STEP 12

Let's rewrite our first equation.Subtracting 6x6x6x from both sides gives us −6x+y=0-6x + y = 0−6x+y=0.

STEP 13

Let's rewrite our second equation.Subtracting 1.5x1.5x1.5x from both sides gives us −1.5x+y=1-1.5x + y = 1−1.5x+y=1.

STEP 14

Looking at the answer choices, option (a) matches our equations perfectly!

STEP 15

The correct answer is (a).

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Start learning now

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What is this asking? Which pair of equations matches the data in the tables for Line A and Line B? Watch out! Don't mix up the lines!
Make sure each equation matches the right table.

1. Find Equation for Line A
2. Find Equation for Line B
3. Match with Answer Choices

Let's peep at Line A's table!
Notice how when $x$ increases by 2, $y$ increases by 12.
That means the slope is $\frac{12}{2} = 6$ .
So, we're looking for an equation that reflects this rate of change!

Let's test this with the first two points.
We have $(1, 6)$ and $(3, 18)$ .
The slope is $\frac{18 - 6}{3 - 1} = \frac{12}{2} = 6$ .
Awesome!

We can use the point-slope form: $y - y_1 = m(x - x_1)$ .
Let's use the point $(1, 6)$ and our slope $m = 6$ .
So, $y - 6 = 6(x - 1)$ .

Simplify! $y - 6 = 6x - 6$ .
Add 6 to both sides to isolate $y$ : $y = 6x$ .

Now, let's investigate Line B!
When $x$ increases by 2, $y$ increases by 3.
The slope is $\frac{3}{2} = 1.5$ .

Let's verify this with $(2, 4)$ and $(4, 7)$ .
The slope is $\frac{7 - 4}{4 - 2} = \frac{3}{2} = 1.5$ .
Perfect!

Again, we use the point-slope form: $y - y_1 = m(x - x_1)$ .
Let's use the point $(2, 4)$ and our slope $m = 1.5$ .
So, $y - 4 = 1.5(x - 2)$ .

Simplify! $y - 4 = 1.5x - 3$ .
Add 4 to both sides to isolate $y$ : $y = 1.5x + 1$ .

Our equations are $y = 6x$ and $y = 1.5x + 1$ .

Let's rewrite our first equation.
Subtracting $6x$ from both sides gives us $-6x + y = 0$ .

Let's rewrite our second equation.
Subtracting $1.5x$ from both sides gives us $-1.5x + y = 1$ .