SAT (Scholastic Assessment Test) Math: Algebra Syllabus

What are the slopes of parallel lines and of perpendicular lines?

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (product = -1).

State slope-intercept form and what each letter means.

y = mx + b, where m is the slope and b is the y-intercept.

State point-slope form of a line.

y - y1 = m(x - x1), using slope m and a point (x1, y1) on the line.

State standard form of a linear equation.

Ax + By = C, where A, B, C are constants (typically integers, A ≥ 0).

How do you find the slope from standard form Ax + By = C?

slope = -A/B.

In the model y = mx + b for a real situation, what does the y-intercept b represent?

The starting/initial value of y when x = 0.

How do you find the x-intercept and the y-intercept of a line?

x-intercept: set y = 0 and solve for x. y-intercept: set x = 0 and solve for y.

In a cost equation C = 0.25m + 40, interpret the intercept 40 and slope 0.25.

40 is the fixed cost (when miles m = 0); 0.25 is the cost per mile.

How do you write the equation of a line given two points?

Find the slope m = (y2-y1)/(x2-x1), then plug m and one point into point-slope form, and simplify to the desired form.

How do you write a line's equation from its graph?

Read the y-intercept b directly off the y-axis, find the slope m from rise/run between two clear points, then write y = mx + b.

Write the equation (slope-intercept) of the line through (0, 3) with slope -2.

y = -2x + 3.

What is the substitution method for solving a system?

Solve one equation for one variable, substitute that expression into the other equation, solve for the remaining variable, then back-substitute.

Solve by substitution: y = 2x and x + y = 9.

x + 2x = 9, so x = 3 and y = 6.

What is the elimination (addition) method for solving a system?

Add or subtract the equations (after scaling so one variable's coefficients are opposites) to eliminate a variable, solve, then back-substitute.

Solve by elimination: x + y = 10 and x - y = 4.

Add: 2x = 14, so x = 7 and y = 3.

When would you scale equations before using elimination?

When no variable's coefficients are already equal or opposite; multiply one or both equations so a variable cancels when added/subtracted.

A system of two linear equations has exactly one solution when?

When the two lines have different slopes (they intersect at one point).

A system of two linear equations has NO solution when?

When the lines are parallel: same slope but different y-intercepts.

A system of two linear equations has infinitely many solutions when?

When the two equations represent the same line (same slope and same y-intercept).

For a system in standard form a1x+b1y=c1 and a2x+b2y=c2, what ratio test gives no solution?

a1/a2 = b1/b2 ≠ c1/c2 (coefficients proportional but constants not) means no solution.

When setting up a system from a word problem (e.g., two items with counts and total cost), what two equations do you typically write?

One equation for the total count/quantity and one for the total value/cost, each in the two unknowns.

How does solving an inequality differ from solving an equation?

It is solved the same way, except you reverse the inequality sign when multiplying or dividing both sides by a negative number.

Solve and graph: -2x > 6.

Divide by -2 and flip: x < -3. Graph: open circle at -3, shade to the left.

When graphing a linear inequality on a number line, when is the endpoint open vs closed?

Open circle for < or > (endpoint not included); closed/filled circle for ≤ or ≥ (endpoint included).

How do you graph a linear inequality in two variables (e.g., y > 2x + 1)?

Graph the boundary line y = 2x + 1 (dashed for < or >, solid for ≤ or ≥), then shade the half-plane that satisfies the inequality.

For y < mx + b vs y > mx + b, which region do you shade?

y < mx + b: shade below the line; y > mx + b: shade above the line.

How do you check which side to shade for a two-variable inequality?

Pick a test point not on the line (often (0,0)); if it makes the inequality true, shade that side, otherwise shade the other side.

What is the solution set of a system of inequalities in two variables?

The overlapping (intersection) region where the shaded areas of all inequalities overlap.

How do you model a real-world constraint like 'at most 20 items' or 'at least $100' with inequalities?

'At most/no more than' uses ≤; 'at least/no less than' uses ≥; combine the constraints (often with x ≥ 0, y ≥ 0) into a system of inequalities.