SAT Math Heart of Algebra focuses on linear equations, linear inequalities, systems, and the connections between algebraic expressions and graphs. Students need this cheat sheet because these questions often reward fast setup, careful interpretation, and efficient equation solving. The same skills appear in word problems, graph questions, table questions, and function notation questions.
A strong command of linear relationships can raise accuracy on both calculator and no-calculator sections.
The most important ideas are slope, intercepts, equivalent forms of linear equations, and solutions to systems. Students should be able to move between forms such as , , and contextual equations. Inequalities follow many of the same rules as equations, except the inequality symbol reverses when multiplying or dividing by a negative number.
For systems, one solution, no solution, and infinitely many solutions can often be identified by comparing slopes and intercepts.
Key Facts
- Slope is the rate of change, and it is calculated by .
- Slope-intercept form is , where is the slope and is the -intercept.
- Standard form can be converted to slope-intercept form by solving for .
- Two lines have one solution when their slopes are different, so .
- Two lines have no solution when they have the same slope but different -intercepts, so and .
- Two lines have infinitely many solutions when they are the same line, so and .
- When solving an inequality, multiplying or dividing by a negative number reverses the sign, such as becoming .
- A linear function has a constant rate of change, meaning equal changes in produce equal changes in .
Vocabulary
- Linear equation
- An equation whose graph is a straight line and whose variables have no exponent greater than .
- Slope
- The ratio of vertical change to horizontal change, written as .
- Y-intercept
- The value of where a line crosses the -axis, which occurs when .
- System of equations
- A set of two or more equations that are solved together by finding values that satisfy all equations.
- Inequality
- A mathematical statement that compares expressions using symbols such as , , , or .
- Rate of change
- The amount one quantity changes for each -unit increase in another quantity.
Common Mistakes to Avoid
- Forgetting to reverse the inequality sign, which is wrong because multiplying or dividing by a negative number changes the order of the values.
- Using the wrong slope formula, which is wrong because must keep the same point order in the numerator and denominator.
- Confusing the -intercept with the -intercept, which is wrong because the -intercept occurs when and the -intercept occurs when .
- Solving only one equation in a system, which is wrong because the solution must satisfy both equations at the same time.
- Ignoring units in word problems, which is wrong because slope, intercepts, and variables often represent real quantities such as dollars per hour or starting fees.
Practice Questions
- 1 A line passes through and . Find its slope and write the equation in the form .
- 2 Solve the system and .
- 3 Solve the inequality and graph the solution on a number line.
- 4 A taxi fare is modeled by , where is the cost in dollars and is the number of miles. Explain what the slope and intercept mean in this context.
Understanding SAT Math Heart of Algebra Question Types
Many SAT algebra problems hide the main relationship inside a story, table, or graph. Start by naming what each variable represents and writing its units. If x is the number of months, then a rate tells how much the quantity changes per month.
If x is miles traveled, the rate may be dollars per mile or gallons per mile. Units help catch mistakes. A starting fee has the same units as the output, not the rate.
For a phone plan, a fixed monthly charge belongs at zero gigabytes. The charge for each gigabyte is the rate. Reading this meaning before calculating prevents many setup errors.
A line can be described in several useful ways, and the best form depends on the task. When a problem gives a starting amount and a repeated change, write the relationship in a form that shows those features clearly. When it gives two points, first find the change in output for a change in input, then use either point to determine the remaining constant.
When it asks where a graph crosses an axis, set the other coordinate to zero. This is more reliable than trying to estimate from a rough graph. Be careful with negative rates.
A negative rate means the output falls as the input rises. It does not mean every output value must be negative.
Systems represent two conditions that must be true at the same time. Their shared solution is often a decision point in a real situation. It may be the number of items at which two pricing plans cost the same, the time when two runners are at the same location, or the values that meet two geometric constraints.
Substitution is often efficient when one equation already gives a variable in terms of the other. Elimination is often efficient when matching coefficients can cancel a variable quickly. After solving, substitute the result into an original equation.
This checks arithmetic and confirms that both conditions are met. If a system has no shared solution, interpret that result in context instead of forcing a numerical answer.
Inequalities describe a range rather than one exact value. On a number line, an open circle excludes an endpoint and a closed circle includes it. In a word problem, words such as at least, no more than, and fewer than determine whether the boundary is included.
A compound inequality may describe values between two limits or values outside a restricted interval. Test a simple value from each region when unsure which side should be shaded. Function notation needs the same careful reading.
A statement about a function at an input asks for the output produced by that input. It does not mean multiplication.
On the SAT, slow down around parentheses, negative signs, and labels on axes. Most wrong answers come from a small interpretation error before the algebra even begins.