SAT Linear Equations: The Complete Guide to Slope, Intercepts, Systems & Inequalities

𝗟𝗜𝗡𝗘𝗔𝗥 𝗘𝗤𝗨𝗔𝗧𝗜𝗢𝗡𝗦 — 𝗧𝗛𝗘 𝗦𝗔𝗧 𝗠𝗔𝗧𝗛 𝗚𝗨𝗜𝗗𝗘 𝗧𝗛𝗔𝗧 𝗧𝗨𝗥𝗡𝗦 𝗟𝗜𝗡𝗘𝗦 𝗜𝗡𝗧𝗢 𝗙𝗥𝗘𝗘 𝗣𝗢𝗜𝗡𝗧𝗦

Linear equations are rarely difficult because of the arithmetic.

The real challenge is recognizing what the equation is telling you.

A question may give you a graph, a table, a word problem, two equations, an inequality, or a strange-looking expression. Underneath all of that, the same small collection of ideas keeps appearing:

• rate of change
• starting value
• slope
• intercepts
• equivalent equations
• systems of equations
• inequalities
• relationships between quantities

Once those patterns become familiar, many apparently different questions become versions of the same problem.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

▰▰▰ 𝟭 ▸ 𝗧𝗛𝗘 𝗧𝗛𝗥𝗘𝗘 𝗠𝗔𝗜𝗡 𝗙𝗢𝗥𝗠𝗦 𝗢𝗙 𝗔 𝗟𝗜𝗡𝗘

A linear equation can appear in several forms. The important skill is knowing what information each form reveals immediately.

𝗦𝗹𝗼𝗽𝗲–𝗶𝗻𝘁𝗲𝗿𝗰𝗲𝗽𝘁 𝗳𝗼𝗿𝗺

𝗒 = 𝗆𝗑 + 𝖻

Here:

𝗆 = slope
𝖻 = y-intercept

The slope tells you how much 𝗒 changes when 𝗑 increases by 1.

The y-intercept tells you the value of 𝗒 when 𝗑 = 0.

For example:

𝗒 = 𝟯𝗑 + 𝟱

The slope is 𝟯.

The y-intercept is 𝟱.

So the line passes through:

(𝟬, 𝟱)

and rises 𝟯 units vertically for every 𝟭 unit of horizontal movement.

𝗦𝘁𝗮𝗻𝗱𝗮𝗿𝗱 𝗳𝗼𝗿𝗺

𝗔𝘅 + 𝗕𝘆 = 𝗖

This form is especially useful when a question involves two quantities contributing to a fixed total.

For example:

𝟰𝘅 + 𝟯𝘆 = 𝟮𝟰

To find the x-intercept, set 𝗒 = 𝟬:

𝟰𝘅 = 𝟮𝟰

𝘅 = 𝟲

So the x-intercept is:

(𝟲, 𝟬)

To find the y-intercept, set 𝗑 = 𝟬:

𝟯𝘆 = 𝟮𝟰

𝘆 = 𝟴

So the y-intercept is:

(𝟬, 𝟴)

If 𝗕 ≠ 𝟬, the slope is:

𝗆 = −𝗔⁄𝗕

𝗣𝗼𝗶𝗻𝘁–𝘀𝗹𝗼𝗽𝗲 𝗳𝗼𝗿𝗺

𝘆 − 𝘆₁ = 𝗆(𝘅 − 𝘅₁)

Use this when you know:

• one point on the line
• the slope

For example, if a line has slope 𝟰 and passes through (𝟮, 𝟯):

𝘆 − 𝟯 = 𝟰(𝘅 − 𝟮)

There is no need to convert this to another form unless the question requires it.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

▰▰▰ 𝟮 ▸ 𝗦𝗟𝗢𝗣𝗘 𝗜𝗦 𝗔 𝗥𝗔𝗧𝗘

One of the most useful ways to understand slope is to attach units to it.

Suppose:

𝗖 = 𝟰𝟱𝗵 + 𝟴𝟬

If 𝗖 is measured in dollars and 𝗵 is measured in hours, then:

𝟰𝟱 = 𝟰𝟱 dollars per hour

and

𝟴𝟬 = 𝟴𝟬 dollars

The equation says:

Starting cost = 𝟴𝟬 dollars

Additional cost = 𝟰𝟱 dollars for every hour

So:

𝗺 = 𝟰𝟱

and

𝖻 = 𝟴𝟬

A useful question to ask whenever you see a linear equation is:

𝗪𝗵𝗮𝘁 𝗱𝗼𝗲𝘀 𝘁𝗵𝗲 𝘀𝗹𝗼𝗽𝗲 𝗺𝗲𝗮𝗻 𝗶𝗻 𝘁𝗵𝗶𝘀 𝗽𝗿𝗼𝗯𝗹𝗲𝗺?

It might represent:

• dollars per month
• miles per hour
• litres per minute
• points per game
• population increase per year
• temperature change per hour

The number is only half of the answer.

The units tell you what the number means.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

▰▰▰ 𝟯 ▸ 𝗙𝗜𝗡𝗗𝗜𝗡𝗚 𝗦𝗟𝗢𝗣𝗘 𝗙𝗥𝗢𝗠 𝗧𝗪𝗢 𝗣𝗢𝗜𝗡𝗧𝗦

If a line passes through:

(𝘅₁, 𝘆₁)

and

(𝘅₂, 𝘆₂)

then:

𝗺 = (𝘆₂ − 𝘆₁)⁄(𝘅₂ − 𝘅₁)

Think:

𝗿𝗶𝘀𝗲⁄𝗿𝘂𝗻

Example:

A line passes through:

(−𝟮, 𝟱)

and

(𝟰, −𝟳)

Then:

𝗺 = (−𝟳 − 𝟱)⁄(𝟰 − (−𝟮))

𝗺 = −𝟭𝟮⁄𝟲

𝗺 = −𝟮

The negative slope means that 𝗒 decreases as 𝗑 increases.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

▰▰▰ 𝟰 ▸ 𝗣𝗔𝗥𝗔𝗟𝗟𝗘𝗟 𝗔𝗡𝗗 𝗣𝗘𝗥𝗣𝗘𝗡𝗗𝗜𝗖𝗨𝗟𝗔𝗥 𝗟𝗜𝗡𝗘𝗦

𝗣𝗮𝗿𝗮𝗹𝗹𝗲𝗹 𝗹𝗶𝗻𝗲𝘀

Parallel nonvertical lines have equal slopes:

𝗺₁ = 𝗺₂

Example:

𝘆 = 𝟮𝘅 + 𝟱

and

𝘆 = 𝟮𝘅 − 𝟴

Both have slope 𝟮, so they are parallel.

𝗣𝗲𝗿𝗽𝗲𝗻𝗱𝗶𝗰𝘂𝗹𝗮𝗿 𝗹𝗶𝗻𝗲𝘀

For two nonvertical perpendicular lines:

𝗺₁𝗺₂ = −𝟭

So if:

𝗺₁ = 𝟯

then:

𝗺₂ = −𝟭⁄𝟯

The quick memory rule is:

𝗳𝗹𝗶𝗽 𝘁𝗵𝗲 𝗳𝗿𝗮𝗰𝘁𝗶𝗼𝗻 𝗮𝗻𝗱 𝗰𝗵𝗮𝗻𝗴𝗲 𝘁𝗵𝗲 𝘀𝗶𝗴𝗻.

There is one important special case.

A horizontal line has slope 𝟬.

A vertical line has undefined slope.

A horizontal line and a vertical line are perpendicular.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

▰▰▰ 𝟱 ▸ 𝗫–𝗜𝗡𝗧𝗘𝗥𝗖𝗘𝗣𝗧 𝗔𝗡𝗗 𝗬–𝗜𝗡𝗧𝗘𝗥𝗖𝗘𝗣𝗧

An intercept is where a graph meets an axis.

To find the x-intercept:

Set:

𝘆 = 𝟬

To find the y-intercept:

Set:

𝘅 = 𝟬

Example:

𝟱𝘅 + 𝟮𝘆 = 𝟮𝟬

For the x-intercept:

𝟱𝘅 = 𝟮𝟬

𝘅 = 𝟰

So:

(𝟰, 𝟬)

For the y-intercept:

𝟮𝘆 = 𝟮𝟬

𝘆 = 𝟭𝟬

So:

(𝟬, 𝟭𝟬)

A common mistake is reporting only 𝟰 or 𝟭𝟬 when the question asks for the coordinate.

Always check exactly what the question requests.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

▰▰▰ 𝟲 ▸ 𝗦𝗬𝗦𝗧𝗘𝗠𝗦 𝗢𝗙 𝗟𝗜𝗡𝗘𝗔𝗥 𝗘𝗤𝗨𝗔𝗧𝗜𝗢𝗡𝗦

A system contains two or more equations that must be true at the same time.

For example:

𝟮𝘅 + 𝘆 = 𝟭𝟭

𝘅 − 𝘆 = 𝟭

Adding the equations gives:

𝟯𝘅 = 𝟭𝟮

so:

𝘅 = 𝟰

Substitute:

𝟰 − 𝘆 = 𝟭

𝘆 = 𝟯

Therefore:

(𝟰, 𝟯)

is the solution.

But you should not automatically solve for both variables.

If the question asks for:

𝘅 + 𝘆

look for a way to obtain that combination directly.

Suppose:

𝟮𝘅 + 𝟯𝘆 = 𝟭𝟮

𝟱𝘅 + 𝟰𝘆 = 𝟮𝟯

The target may sometimes be obtained by multiplying and adding equations rather than finding 𝘅 and 𝘆 separately.

𝗦𝗔𝗧 𝘁𝗶𝗺𝗲 𝘁𝗶𝗽:

𝗙𝗶𝗿𝘀𝘁 𝗿𝗲𝗮𝗱 𝘁𝗵𝗲 𝘁𝗮𝗿𝗴𝗲𝘁.

Then decide which algebraic operation reaches that target most directly.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

▰▰▰ 𝟳 ▸ 𝗢𝗡𝗘 𝗦𝗢𝗟𝗨𝗧𝗜𝗢𝗡, 𝗡𝗢 𝗦𝗢𝗟𝗨𝗧𝗜𝗢𝗡, 𝗢𝗥 𝗜𝗡𝗙𝗜𝗡𝗜𝗧𝗘𝗟𝗬 𝗠𝗔𝗡𝗬?

Two linear equations can have:

𝟭. exactly one solution
𝟮. no solution
𝟯. infinitely many solutions

Think about the graphs.

𝗢𝗻𝗲 𝘀𝗼𝗹𝘂𝘁𝗶𝗼𝗻

The lines intersect at one point.

𝗡𝗼 𝘀𝗼𝗹𝘂𝘁𝗶𝗼𝗻

The lines are distinct and parallel.

𝗜𝗻𝗳𝗶𝗻𝗶𝘁𝗲𝗹𝘆 𝗺𝗮𝗻𝘆 𝘀𝗼𝗹𝘂𝘁𝗶𝗼𝗻𝘀

The two equations represent the same line.

For equations:

𝗔₁𝘅 + 𝗕₁𝘆 = 𝗖₁

𝗔₂𝘅 + 𝗕₂𝘆 = 𝗖₂

a particularly useful test is to check whether one entire equation is a constant multiple of the other.

Example:

𝟲𝘅 + 𝗸𝘆 = 𝟵

𝟮𝘅 + 𝟱𝘆 = 𝟯

For infinitely many solutions, the first equation must be exactly 𝟯 times the second.

Since:

𝟯(𝟮𝘅 + 𝟱𝘆) = 𝟲𝘅 + 𝟭𝟱𝘆

we need:

𝗸 = 𝟭𝟱

The constants also agree:

𝟯(𝟯) = 𝟵

So:

𝗸 = 𝟭𝟱

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

▰▰▰ 𝟴 ▸ 𝗟𝗜𝗡𝗘𝗔𝗥 𝗜𝗡𝗘𝗤𝗨𝗔𝗟𝗜𝗧𝗜𝗘𝗦

Most of the algebra rules remain the same.

There is one rule you cannot forget:

⚠️ 𝗪𝗵𝗲𝗻 𝘆𝗼𝘂 𝗺𝘂𝗹𝘁𝗶𝗽𝗹𝘆 𝗼𝗿 𝗱𝗶𝘃𝗶𝗱𝗲 𝗯𝘆 𝗮 𝗻𝗲𝗴𝗮𝘁𝗶𝘃𝗲 𝗻𝘂𝗺𝗯𝗲𝗿, 𝗿𝗲𝘃𝗲𝗿𝘀𝗲 𝘁𝗵𝗲 𝗶𝗻𝗲𝗾𝘂𝗮𝗹𝗶𝘁𝘆 𝘀𝗶𝗴𝗻.

Example:

−𝟯𝘅 > 𝟭𝟮

Divide by −𝟯:

𝘅 < −𝟰

Notice:

became <

That reversal is essential.

𝗚𝗿𝗮𝗽𝗵𝗶𝗻𝗴 𝗶𝗻𝗲𝗾𝘂𝗮𝗹𝗶𝘁𝗶𝗲𝘀

Use:

≤ or ≥ → solid boundary

< or > → dashed boundary

For a two-variable inequality, choose a convenient test point that is not on the boundary.

The origin (𝟬, 𝟬) is often convenient, but it is not mandatory. If the origin lies on the boundary, choose another point.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

▰▰▰ 𝟵 ▸ 𝗧𝗨𝗥𝗡𝗜𝗡𝗚 𝗪𝗢𝗥𝗗𝗦 𝗜𝗡𝗧𝗢 𝗘𝗤𝗨𝗔𝗧𝗜𝗢𝗡𝗦

Many linear-equation questions are really translation problems.

Watch for these phrases:

𝗶𝘀 → =

𝗽𝗲𝗿 → rate or multiplication

𝗲𝗮𝗰𝗵 → rate or multiplication

𝗶𝗻𝗰𝗿𝗲𝗮𝘀𝗲𝗱 𝗯𝘆 → +

𝗱𝗲𝗰𝗿𝗲𝗮𝘀𝗲𝗱 𝗯𝘆 → −

𝗺𝗼𝗿𝗲 𝘁𝗵𝗮𝗻 → +

𝗮𝘁 𝗹𝗲𝗮𝘀𝘁 → ≥

𝗮𝘁 𝗺𝗼𝘀𝘁 → ≤

𝗹𝗲𝘀𝘀 𝘁𝗵𝗮𝗻 → <

𝗴𝗿𝗲𝗮𝘁𝗲𝗿 𝘁𝗵𝗮𝗻 → >

𝗼𝗳 → multiplication

But do not translate mechanically.

For example:

“𝟱 less than 𝘅”

means:

𝘅 − 𝟱

while:

“𝘅 less than 𝟱”

means:

𝟱 − 𝘅

The order matters.

𝗗𝗲𝗳𝗶𝗻𝗲 𝘆𝗼𝘂𝗿 𝘃𝗮𝗿𝗶𝗮𝗯𝗹𝗲

If:

𝘅 = number of small boxes

then every later appearance of 𝘅 has a clear meaning.

This simple habit prevents you from solving an equation correctly and then reporting the wrong quantity.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

▰▰▰ 𝟭𝟬 ▸ 𝗪𝗢𝗥𝗗 𝗣𝗥𝗢𝗕𝗟𝗘𝗠𝗦: 𝗙𝗜𝗡𝗗 𝗧𝗛𝗘 𝗥𝗔𝗧𝗘 𝗔𝗡𝗗 𝗧𝗛𝗘 𝗦𝗧𝗔𝗥𝗧𝗜𝗡𝗚 𝗩𝗔𝗟𝗨𝗘

Suppose a gym charges:

₹𝟭,𝟮𝟬𝟬 to join

and

₹𝟳𝟬𝟬 each month.

Let:

𝗺 = number of months

Then:

𝗖 = 𝟳𝟬𝟬𝗺 + 𝟭,𝟮𝟬𝟬

The 𝟳𝟬𝟬 is the monthly rate.

The ₹𝟭,𝟮𝟬𝟬 is the starting charge.

After 𝟵 months:

𝗖 = 𝟳𝟬𝟬(𝟵) + 𝟭,𝟮𝟬𝟬

𝗖 = 𝟲,𝟯𝟬𝟬 + 𝟭,𝟮𝟬𝟬

𝗖 = ₹𝟳,𝟱𝟬𝟬

A useful mental picture is:

𝗧𝗼𝘁𝗮𝗹 = 𝗶𝗻𝗶𝘁𝗶𝗮𝗹 𝗮𝗺𝗼𝘂𝗻𝘁 + (𝗿𝗮𝘁𝗲 × 𝗻𝘂𝗺𝗯𝗲𝗿 𝗼𝗳 𝘂𝗻𝗶𝘁𝘀)

This pattern appears constantly in real-world linear models.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

▰▰▰ 𝟭𝟭 ▸ 𝗗𝗜𝗦𝗧𝗥𝗜𝗕𝗨𝗧𝗜𝗡𝗚 𝗔𝗡𝗗 𝗖𝗢𝗠𝗕𝗜𝗡𝗜𝗡𝗚 𝗧𝗘𝗥𝗠𝗦

Consider:

𝟯(𝘅 − 𝟰) = 𝟱𝘅 + 𝟴

Distribute first:

𝟯𝘅 − 𝟭𝟮 = 𝟱𝘅 + 𝟴

Move the variable terms:

−𝟮𝟬 = 𝟮𝘅

Therefore:

𝘅 = −𝟭𝟬

A common error is distributing a negative incorrectly.

For example:

−𝟮(𝘅 − 𝟱)

must become:

−𝟮𝘅 + 𝟭𝟬

because:

−𝟮 × 𝘅 = −𝟮𝘅

and

−𝟮 × (−𝟱) = +𝟭𝟬

The negative sign must affect every term inside the parentheses.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

▰▰▰ 𝟭𝟮 ▸ 𝗧𝗛𝗘 𝗧𝗔𝗥𝗚𝗘𝗧 𝗜𝗦 𝗡𝗢𝗧 𝗔𝗟𝗪𝗔𝗬𝗦 𝗫

This is one of the most useful habits for timed questions.

Suppose you are given:

𝟮𝘅 + 𝟱𝘆 = 𝟭𝟵

𝟱𝘅 + 𝟮𝘆 = 𝟭𝟲

and asked for:

𝘅 + 𝘆

Add the equations:

𝟮𝘅 + 𝟱𝘆 + 𝟱𝘅 + 𝟮𝘆 = 𝟭𝟵 + 𝟭𝟲

𝟳𝘅 + 𝟳𝘆 = 𝟯𝟱

Factor:

𝟳(𝘅 + 𝘆) = 𝟯𝟱

Therefore:

𝘅 + 𝘆 = 𝟱

There was no reason to find 𝘅 and 𝘆 separately.

𝗟𝗼𝗼𝗸 𝗮𝘁 𝘁𝗵𝗲 𝗾𝘂𝗲𝘀𝘁𝗶𝗼𝗻 𝗳𝗶𝗿𝘀𝘁.

Then choose the algebra that produces exactly what is requested.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

▰▰▰ 𝟭𝟯 ▸ 𝗖𝗢𝗠𝗠𝗢𝗡 𝗟𝗜𝗡𝗘𝗔𝗥 𝗘𝗤𝗨𝗔𝗧𝗜𝗢𝗡 𝗧𝗥𝗔𝗣𝗦

𝗧𝗿𝗮𝗽 𝟭: 𝗙𝗼𝗿𝗴𝗲𝘁𝘁𝗶𝗻𝗴 𝘁𝗵𝗲 𝗾𝘂𝗲𝘀𝘁𝗶𝗼𝗻

You solve for 𝘅.

The question asks for:

𝟮𝘅 + 𝟭

Your algebra may be perfect, but your final answer is still wrong if you stop too early.

𝗧𝗿𝗮𝗽 𝟮: 𝗙𝗼𝗿𝗴𝗲𝘁𝘁𝗶𝗻𝗴 𝘁𝗵𝗲 𝗻𝗲𝗴𝗮𝘁𝗶𝘃𝗲

Dividing an inequality by a negative reverses the sign.

𝗧𝗿𝗮𝗽 𝟯: 𝗪𝗿𝗼𝗻𝗴 𝘀𝗹𝗼𝗽𝗲

For:

𝗔𝘅 + 𝗕𝘆 = 𝗖

the slope is:

−𝗔⁄𝗕

not:

𝗔⁄𝗕

𝗧𝗿𝗮𝗽 𝟰: 𝗨𝗻𝗶𝘁 𝗰𝗼𝗻𝗳𝘂𝘀𝗶𝗼𝗻

If one quantity is measured in minutes and another in hours, convert before interpreting the rate.

Likewise, do not mix cents and dollars without converting.

𝗧𝗿𝗮𝗽 𝟱: 𝗜𝗻𝘁𝗲𝗿𝗰𝗲𝗽𝘁 𝗰𝗼𝗻𝗳𝘂𝘀𝗶𝗼𝗻

The x-intercept has:

𝘆 = 𝟬

The y-intercept has:

𝘅 = 𝟬

𝗧𝗿𝗮𝗽 𝟲: 𝗦𝗶𝗴𝗻 𝗲𝗿𝗿𝗼𝗿𝘀 𝗶𝗻 𝗽𝗮𝗿𝗲𝗻𝘁𝗵𝗲𝘀𝗲𝘀

−𝟯(𝘅 − 𝟰)

becomes:

−𝟯𝘅 + 𝟭𝟮

not:

−𝟯𝘅 − 𝟭𝟮

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

▰▰▰ 𝟭𝟰 ▸ 𝗪𝗛𝗘𝗡 𝗔𝗟𝗚𝗘𝗕𝗥𝗔 𝗜𝗦 𝗡𝗢𝗧 𝗧𝗛𝗘 𝗙𝗔𝗦𝗧𝗘𝗦𝗧 𝗠𝗘𝗧𝗛𝗢𝗗

Sometimes the answer choices themselves can save time.

Suppose the question asks for a numerical value and every answer choice is a possible value of 𝘅.

Instead of doing several lines of algebra, you can substitute an answer choice into the original equation.

If one choice makes the equation true, you have found the solution.

This is especially useful when:

• the algebra is unusually long
• the answer choices are simple numbers
• substitution is quick
• you can eliminate several choices immediately

The important rule is:

𝗖𝗵𝗲𝗰𝗸 𝘁𝗵𝗲 𝗼𝗿𝗶𝗴𝗶𝗻𝗮𝗹 𝗰𝗼𝗻𝗱𝗶𝘁𝗶𝗼𝗻, not an altered version that may contain an algebra mistake.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

▰▰▰ 𝟭𝟱 ▸ 𝗣𝗥𝗔𝗖𝗧𝗜𝗖𝗘 𝗦𝗘𝗧

𝟭

A line passes through:

(−𝟮, 𝟱)

and

(𝟰, −𝟳)

What is the slope?

𝟮

Solve:

𝟯(𝘅 − 𝟰) = 𝟱𝘅 + 𝟴

𝟯

For:

𝟰𝘅 + 𝟯𝘆 = 𝟭𝟮

what is the x-intercept?

𝟰

For what value of 𝗰 do the equations

𝟴𝘅 − 𝟮𝘆 = 𝟳

and

𝟰𝘅 + 𝗰𝘆 = 𝟭𝟭

have no solution?

𝟱

A gym charges ₹𝟭,𝟮𝟬𝟬 to join and ₹𝟳𝟬𝟬 each month.

Write an equation for the total cost 𝗖 after 𝗺 months.

Then find the cost after 𝟵 months.

𝟲

If:

𝟮𝘅 + 𝟱𝘆 = 𝟭𝟵

and

𝟱𝘅 + 𝟮𝘆 = 𝟭𝟲

what is:

𝘅 + 𝘆?

𝟳

Solve:

−𝟰𝘅 + 𝟵 ≤ 𝟮𝟱

𝟴

A line has slope −𝟯 and passes through (𝟮, 𝟱).

Write its equation in point-slope form.

𝟵

Two lines have slopes:

𝟮⁄𝟱

and

−𝟱⁄𝟮

What is the relationship between the lines?

𝟭𝟬

A quantity is represented by:

𝗣 = 𝟭𝟱𝘁 + 𝟰𝟬

What does the number 𝟭𝟱 represent if 𝗣 is measured in dollars and 𝘁 is measured in hours?

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

▰▰▰ 𝗔𝗡𝗦𝗪𝗘𝗥𝗦

𝟭

𝗺 = (−𝟳 − 𝟱)⁄(𝟰 − (−𝟮))

𝗺 = −𝟭𝟮⁄𝟲

𝗺 = −𝟮

𝟮

𝟯𝘅 − 𝟭𝟮 = 𝟱𝘅 + 𝟴

−𝟮𝟬 = 𝟮𝘅

𝘅 = −𝟭𝟬

𝟯

Set:

𝘆 = 𝟬

Then:

𝟰𝘅 = 𝟭𝟮

𝘅 = 𝟯

The x-intercept is:

(𝟯, 𝟬)

𝟰

For no solution, the lines must have the same slope but different intercepts.

The first equation has slope:

−𝟴⁄(−𝟮) = 𝟰

The second equation has slope:

−𝟰⁄𝗰

Set:

−𝟰⁄𝗰 = 𝟰

−𝟰 = 𝟰𝗰

𝗰 = −𝟭

With 𝗰 = −𝟭, the two equations have equal slopes but do not represent the same line, so there is no solution.

𝟱

𝗖 = 𝟳𝟬𝟬𝗺 + 𝟭,𝟮𝟬𝟬

At 𝗺 = 𝟵:

𝗖 = 𝟳𝟬𝟬(𝟵) + 𝟭,𝟮𝟬𝟬

𝗖 = ₹𝟳,𝟱𝟬𝟬

𝟲

Add the equations:

𝟳𝘅 + 𝟳𝘆 = 𝟯𝟱

Therefore:

𝘅 + 𝘆 = 𝟱

𝟳

−𝟰𝘅 + 𝟵 ≤ 𝟮𝟱

−𝟰𝘅 ≤ 𝟭𝟲

Divide by −𝟰 and reverse the sign:

𝘅 ≥ −𝟰

𝟴

𝘆 − 𝟱 = −𝟯(𝘅 − 𝟮)

𝟵

The product of the slopes is:

(𝟮⁄𝟱)(−𝟱⁄𝟮) = −𝟭

Therefore the lines are perpendicular.

𝟭𝟬

𝟭𝟱 represents a rate of:

𝗱𝗼𝗹𝗹𝗮𝗿𝘀 𝗽𝗲𝗿 𝗵𝗼𝘂𝗿

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

▰▰▰ 𝗧𝗛𝗘 𝟯𝟬-𝗦𝗘𝗖𝗢𝗡𝗗 𝗟𝗜𝗡𝗘𝗔𝗥 𝗘𝗤𝗨𝗔𝗧𝗜𝗢𝗡 𝗖𝗛𝗘𝗖𝗞

Before submitting an answer, ask:

☐ What exactly is the question asking me to find?

☐ What does each variable represent?

☐ What are the units?

☐ Is the slope positive, negative, zero, or undefined?

☐ Did I use the correct intercept?

☐ If I divided an inequality by a negative number, did I reverse the sign?

☐ If I am solving a system, do I really need both variables?

☐ Does my final value make sense in the context?

☐ Did I answer with the requested quantity rather than an intermediate value?

Linear equations become much easier when you stop treating every question as a fresh problem.

Look for the structure.

𝗥𝗮𝘁𝗲 → 𝘀𝗹𝗼𝗽𝗲

𝗦𝘁𝗮𝗿𝘁𝗶𝗻𝗴 𝘃𝗮𝗹𝘂𝗲 → 𝗶𝗻𝘁𝗲𝗿𝗰𝗲𝗽𝘁

𝗠𝗲𝗲𝘁𝘀 𝘁𝗵𝗲 𝘅-𝗮𝘅𝗶𝘀 → 𝘆 = 𝟬

𝗠𝗲𝗲𝘁𝘀 𝘁𝗵𝗲 𝘆-𝗮𝘅𝗶𝘀 → 𝘅 = 𝟬

𝗣𝗮𝗿𝗮𝗹𝗹𝗲𝗹 → 𝘀𝗮𝗺𝗲 𝘀𝗹𝗼𝗽𝗲

𝗣𝗲𝗿𝗽𝗲𝗻𝗱𝗶𝗰𝘂𝗹𝗮𝗿 → 𝗻𝗲𝗴𝗮𝘁𝗶𝘃𝗲 𝗿𝗲𝗰𝗶𝗽𝗿𝗼𝗰𝗮𝗹 𝘀𝗹𝗼𝗽𝗲𝘀

𝗡𝗲𝗴𝗮𝘁𝗶𝘃𝗲 𝗱𝗶𝘃𝗶𝘀𝗶𝗼𝗻 → 𝗳𝗹𝗶𝗽 𝘁𝗵𝗲 𝗶𝗻𝗲𝗾𝘂𝗮𝗹𝗶𝘁𝘆

𝗦𝘆𝘀𝘁𝗲𝗺 → 𝗹𝗼𝗼𝗸 𝗳𝗼𝗿 𝘁𝗵𝗲 𝗿𝗲𝗾𝘂𝗶𝗿𝗲𝗱 𝗰𝗼𝗺𝗯𝗶𝗻𝗮𝘁𝗶𝗼𝗻 𝗯𝗲𝗳𝗼𝗿𝗲 𝘀𝗼𝗹𝘃𝗶𝗻𝗴

The objective is not to perform more algebra.

The objective is to recognize the shortest correct path to the answer.
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━