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2. Equations & Inequalities in Two Variables
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United States of AmericaTennessee
Algebra 1

2.02 Standard form

Lesson
Practice
Lesson

Concept summary

The standard form of a linear relationship is a way of writing the equation with all of the variables on one side:

\displaystyle Ax+By=C
\bm{A}
is a non-negative integer
\bm{B,C}
are integers
\bm{A,B}
are not both 0

To draw the graph from standard form, we can find and plot the x and y-intercepts or convert to slope-intercept form.

The standard form is helpful when looking at scenarios that have a mixture of two different items.

When we identify the intercepts in a mixture scenario, it can be interpreted as the amount of that item when none of the other item is included.

Worked examples

Example 1

Draw the graph of the line 5x-3y=-15 on the coordinate plane.

Approach

We can find both the x and y-intercept and then graph the line using those two points.

Solution

Find the x-intercept by setting y=0 and solving:

\displaystyle 5x-3y\displaystyle =\displaystyle -15State the given equation
\displaystyle 5x-3\left(0\right)\displaystyle =\displaystyle -15Set y=0
\displaystyle 5x\displaystyle =\displaystyle -15Simplify
\displaystyle x\displaystyle =\displaystyle -3Divide both sides by 5

Find the y-intercept by setting x=0 and solving:

\displaystyle 5x-3y\displaystyle =\displaystyle -15State the given equation
\displaystyle 5\left(0\right)-3y\displaystyle =\displaystyle -15Set x=0
\displaystyle -3y\displaystyle =\displaystyle -15Simplify
\displaystyle y\displaystyle =\displaystyle 5Divide both sides by -3
-5
-4
-3
-2
-1
1
2
3
x
-1
1
2
3
4
5
6
7
y

Reflection

We could have also converted to slope-intercept and graphed using the slope and y-intercept. We can decide unless the question specifies how to do it.

If the x and y-intercept happened to be same point, \left(0,0\right), then we need to find another point by substituting another x-value into the equation and solving for y.

Example 2

Darius wants to buy a mix of garlic and chipotle powders for seasoning tacos. Garlic powder costs \$ 4/\text{lb}. Chipotle powder costs \$ 7/\text{lb}.

Darius spends exactly \$ 14 on spices.

Let x represent the amount of garlic powder Darius buys and let y represent the amount of chipotle powder he buys. Write an equation to represent this scenario.

Approach

The cost of each spice will be the cost per pound multiplied by the amount purchased.

The total cost will be the sum of the two spice costs and is equal to \$14.

Solution

Cost of just the garlic: 4 \cdot x

Cost of just the chipotle: 7 \cdot y

Total cost: 4x+7y=14

Example 3

A tour company travels to the Great Smoky Mountains National Park. They use a combination of buses and vans to get tourists to their destination. One bus can take 42 passengers and one van can take 7 passengers. One day they have 168 people are register for the tour.

a

Write an equation in standard from that could be used to model the number of buses and vans they could use to transport all the people registered.

Approach

In words, we can start with the idea that: \left(\text{Number of people on buses}\right)+\left(\text{Number of people on vans}\right)=\text{Total number of people} and that: \text{Number of people on buses}=42 \cdot \left(\text{Number of buses}\right) and:\text{Number of people on vans}=7 \cdot \left(\text{Number of vans}\right)

We will then need to declare variables and write an equation using them.

Solution

Let b represent the number of buses used and let v represent the number of vans used.

So the \text{Number of people on buses}=42b and \text{Number of people on vans}=7v

Which finally gives us the whole equation: 42b+7v=168

Reflection

It is important to declare variables and it can be helpful to use variables that relate to the quantities in the scenario to make sure we don't mix them up.

b

Graph the equation with an appropriate scale and labels.

Approach

In this case there isn't a clear independent and dependent variable, so we can put b on the horizontal axis and v as on the vertical axis.

To determine an appropriate scale we can first find the values of the intercepts as those can give an idea of the maximum values for each axis.

Solution

Find the b-intercept:

\displaystyle 42b+7v\displaystyle =\displaystyle 168Equation from part (a)
\displaystyle 42b+7(0)\displaystyle =\displaystyle 168Substitute v=0 to find b-intercept
\displaystyle 42b\displaystyle =\displaystyle 168Evaluate the product
\displaystyle b\displaystyle =\displaystyle 4Division property of equality

Find the v-intercept:

\displaystyle 42b+7v\displaystyle =\displaystyle 168Equation from part (a)
\displaystyle 42(0)+7v\displaystyle =\displaystyle 168Substitute b=0 to find v-intercept
\displaystyle 7v\displaystyle =\displaystyle 168Evaluate the product
\displaystyle v\displaystyle =\displaystyle 24Division property of equality
1
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4
5
\text{Number of buses }\left(b\right)
2
4
6
8
10
12
14
16
18
20
22
24
26
\text{Number of vans }\left(v\right)

An appropriate scale would be going up by 1 along the b-axis to a maximum of 5 and going up by 2 or 4 along the v-axis to a maximum of 26.

Reflection

In this case, if we made b the independent variable instead, the graph would could like:

4
8
12
16
20
24
\text{Number of vans }\left(v\right)
1
2
3
4
5
\text{Number of buses }\left(b\right)
c

Predict the number of vans that would be required if only 1 bus was available.

Approach

We can find the point along the b-axis where b=1 and then up to the line and across to the v-axis to find the corresponding value for v, the number of vans.

Solution

1
2
3
4
5
\text{Number of buses }\left(b\right)
2
4
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10
12
14
16
18
20
22
24
26
\text{Number of vans }\left(v\right)

Using the graph, we can see that if only 1 bus was available, they would need 18 vans.

Reflection

We can check using the equation.

\displaystyle 42b+7v\displaystyle =\displaystyle 168Equation from part (a)
\displaystyle 42(1)+7v\displaystyle =\displaystyle 168Substitute in for 1 bus
\displaystyle 42+7v\displaystyle =\displaystyle 168Evaluate the product
\displaystyle 7v\displaystyle =\displaystyle 126Subtraction property of equality
\displaystyle v\displaystyle =\displaystyle 18Division property of equality

Outcomes

A1.N.Q.A.1

Use units as a way to understand real-world problems.*

A1.N.Q.A.1.A

Choose and interpret the scale and the origin in graphs and data displays.*

A1.A.CED.A.2

Create equations in two variables to represent relationships between quantities and use them to solve problems in a real-world context. Graph equations with two variables on coordinate axes with labels and scales, and use the graphs to make predictions.*

A1.A.CED.A.3

Create individual and systems of equations and/or inequalities to represent constraints in a contextual situation, and interpret solutions as viable or non-viable.*

A1.A.REI.D.5

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

A1.MP1

Make sense of problems and persevere in solving them.

A1.MP2

Reason abstractly and quantitatively.

A1.MP3

Construct viable arguments and critique the reasoning of others.

A1.MP4

Model with mathematics.

A1.MP5

Use appropriate tools strategically.

A1.MP6

Attend to precision.

A1.MP7

Look for and make use of structure.

A1.MP8

Look for and express regularity in repeated reasoning.

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