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2. Equations & Inequalities in Two Variables
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United States of AmericaFL
Grade 912

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

Outcomes

MA.912.AR.2.1

Given a real-world context, write and solve one-variable multi-step linear equations.

MA.912.AR.2.2

Write a linear two-variable equation to represent the relationship between two quantities from a graph, a written description or a table of values within a mathematical or real-world context.

MA.912.AR.2.4

Given a table, equation or written description of a linear function, graph that function, and determine and interpret its key features.

MA.912.AR.2.5

Solve and graph mathematical and real-world problems that are modeled with linear functions. Interpret key features and determine constraints in terms of the context.

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