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Equations With
Two Variables

Master the relationship between \(x\) and \(y\) in linear equations: graphing, solving, and interpreting solutions as ordered pairs.

SAT Math

What Is an Equation in Two Variables?

An equation in two variables contains two different unknowns, most often \(x\) and \(y\). A solution is any ordered pair \((x, y)\) that makes the equation true.

Solutions Live on the Coordinate Plane

x y 1 2 3 4 5 1 2 3 (2, 3) (3, 2) (5, 0) \(x + y = 5\)

Each ordered pair \((x, y)\) that satisfies the equation is a point on the line. The graph of a linear equation in two variables is always a straight line.

Why This Matters on the SAT

Algebra
The SAT tests whether you can solve for one variable given a relationship between two.
Graphing
Many questions ask you to identify which graph matches an equation or vice versa.
Word Problems
Real-world scenarios translate into two-variable equations you must set up and interpret.

Standard Form: \(Ax + By = C\)

A linear equation in two variables can be written in standard form:

\(x\)-intercept
Set \(y = 0\): the point \(\left(\frac{C}{A},\, 0\right)\)
\(y\)-intercept
Set \(x = 0\): the point \(\left(0,\, \frac{C}{B}\right)\)

Slope-Intercept Form: \(y = mx + b\)

Rearranging a two-variable equation to isolate \(y\) reveals the slope \(m\) and the \(y\)-intercept \(b\) directly.

x y (0, 1) = b run = 1 rise = 2 (1, 3) \(y = 2x + 1\)

Standard vs Slope-Intercept

Standard Form Slope-Intercept Form
Format \(Ax + By = C\) \(y = mx + b\)
Best for Finding intercepts quickly Reading slope and \(y\)-intercept
SAT usage Systems, integer coefficients Graphing, interpreting rate of change
Convert Solve for \(y\) to get slope-intercept Multiply through to clear fractions

How to Solve Two-Variable Problems

1
Identify the equation form. Is it standard \(Ax + By = C\) or slope-intercept \(y = mx + b\)?
2
Determine what is asked. A specific solution? The slope? An intercept? A graph?
3
Substitute or rearrange. Plug in the given value and solve for the remaining unknown.
4
Check your answer. Substitute the ordered pair back into the original equation to verify.

Example 1: Finding a Solution

Problem: If \(3x + 2y = 18\), what is the value of \(y\) when \(x = 4\)?

1
Substitute \(x = 4\) into the equation: \(3(4) + 2y = 18\)
2
Simplify: \(12 + 2y = 18\)
3
Solve: \(2y = 6\), so \(y = 3\)
4
Verify: \(3(4) + 2(3) = 12 + 6 = 18\) ✓

Example 2: Converting to Slope-Intercept

Problem: Rewrite \(5x - 2y = 10\) in slope-intercept form and identify the slope and \(y\)-intercept.

1
Isolate the \(y\)-term: \(-2y = -5x + 10\)
2
Divide by \(-2\): \(y = \frac{5}{2}x - 5\)
3
Read off: slope \(m = \frac{5}{2}\), \(y\)-intercept \(b = -5\)

Example 3: A Word Problem

Problem: A store sells pens for $2 each and notebooks for $5 each. A customer spends exactly $30. If the customer buys \(p\) pens and \(n\) notebooks, which equation models this, and how many pens if \(n = 4\)?

1
Write the equation: \(2p + 5n = 30\)
2
Substitute \(n = 4\): \(2p + 5(4) = 30\)
3
Simplify: \(2p + 20 = 30\), so \(2p = 10\), thus \(p = 5\)
4
Verify: \(2(5) + 5(4) = 10 + 20 = 30\) ✓

Interpreting Slope in Context

On the SAT, you may be asked what the slope or intercept means in a given situation. The slope is always a rate of change.

\(m\) (slope)
The change in \(y\) for each 1-unit increase in \(x\). In context: "for each additional [x-unit], the [y-quantity] changes by \(m\)."
\(b\) (y-intercept)
The value of \(y\) when \(x = 0\). In context: "the initial value" or "the starting amount."

When Two Equations Meet

x y (2, 3) \(y = x + 1\) \(y = -x + 5\)

When two linear equations share a common solution, their lines intersect. The intersection point \((2, 3)\) satisfies both equations simultaneously: this is the basis for systems of equations.

Checking If a Point Is a Solution

A common SAT question type: "Which ordered pair satisfies the equation?" The method is straightforward:

1
Substitute the \(x\)-value and \(y\)-value from the ordered pair into the equation.
2
Simplify both sides of the equation.
3
Compare: If both sides are equal, the point is a solution. If not, it is not.

Common Mistakes to Avoid

✕ Sign Errors
Forgetting to distribute a negative when moving terms: \(-2y = -5x + 10\) divided by \(-2\) gives \(y = \frac{5}{2}x - 5\), not \(+5\).
✕ Swapping \(x\) and \(y\)
Writing the solution as \((y, x)\) instead of \((x, y)\). Order matters: the first coordinate is always \(x\).
✕ Misreading Slope
Confusing the coefficient of \(x\) in standard form with the slope. In \(Ax + By = C\), the slope is \(-\frac{A}{B}\), not \(A\).

Summary of Key Facts

Concept Key Formula What It Tells You
Standard Form \(Ax + By = C\) Intercepts easily
Slope-Intercept \(y = mx + b\) Slope and \(y\)-intercept
Slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\) Rate of change
Solution \((x, y)\) satisfies eq. Point on the line
Intersection Solve both equations Common solution

Question 1 of 5

If \(2x + 3y = 12\), what is the value of \(y\) when \(x = 3\)?

Question 2 of 5

Which ordered pair is a solution to \(3x + y = 10\)?

Question 3 of 5

The equation \(6x - 2y = 18\) is rewritten in slope-intercept form. What is the slope?

Question 4 of 5

A gym charges a one-time fee of $25 and $10 per month. If \(C\) is the total cost after \(m\) months, which equation models this relationship?

Question 5 of 5

What is the \(x\)-intercept of the line \(3x + 4y = 24\)?