9.2 Systems of Equations and Inequalities

Why Systems and Inequalities Appear on the USTET

After single linear equations, the USTET Mathematics subtest expects you to handle two equations with two unknowns and to reason with inequalities. These skills also appear in Senior High School General Mathematics, so expect both "solve the system" stems and short applied problems (ticket prices, mixture of two coin types, age relationships). Under timed conditions, pick the method that keeps arithmetic small: substitution when one variable is already isolated, elimination when coefficients match or are easy multiples.

Systems of Linear Equations

A system is a set of equations that must be true at the same time. For two lines in the plane:

Most USTET items are written so there is exactly one solution.

Method 1: Substitution

Use when one equation is already solved for a variable, or is easy to rearrange.

Solve:

y = 2x − 1 3x + y = 14

Step 1. Substitute y from the first equation into the second: 3x + (2x − 1) = 14. Step 2. Combine: 5x − 1 = 14 → 5x = 15 → x = 3. Step 3. Back-substitute: y = 2(3) − 1 = 5. Solution: (3, 5).

Method 2: Elimination

Align like terms and add (or subtract) to cancel one variable.

Solve:

2x + 3y = 12 4x − 3y = 6

Add the equations: (2x + 4x) + (3y − 3y) = 12 + 6 → 6x = 18 → x = 3. Plug into the first: 2(3) + 3y = 12 → 6 + 3y = 12 → 3y = 6 → y = 2. Solution: (3, 2).

If coefficients are not ready to cancel, multiply one or both equations first. To eliminate x in

3x + 2y = 11 5x − y = 7

multiply the second equation by 2: 10x − 2y = 14. Add to the first: 13x = 25 → x = 25/13. Expect some USTET choices as fractions in lowest terms.

Worked Example: Peso Word System

Adult tickets cost PHP 150 and student tickets cost PHP 90. A group buys 8 tickets for PHP 960. How many adult tickets were bought?

Let a = adult tickets and s = student tickets.

a + s = 8 150a + 90s = 960

From the first, s = 8 − a. Substitute: 150a + 90(8 − a) = 960 → 150a + 720 − 90a = 960 → 60a = 240 → a = 4. Then s = 4. Check: 4(150) + 4(90) = 600 + 360 = 960.

Linear Inequalities

An inequality compares two expressions with <, >, ≤, or ≥. Solving looks like equation-solving except one critical rule: if you multiply or divide both sides by a negative number, reverse the inequality symbol.

Worked Example: Flip Rule

Solve: −3x + 5 ≥ 20

Subtract 5: −3x ≥ 15. Divide by −3 and reverse: x ≤ −5.

Number-line sketch: closed circle at −5 (because ≤ includes equality), shade leftward.

Compound Inequalities

A compound inequality such as −2 < 3x + 1 ≤ 10 can be split into two pieces solved together:

−2 < 3x + 1 and 3x + 1 ≤ 10.

Subtract 1 throughout: −3 < 3x ≤ 9. Divide by 3: −1 < x ≤ 3. So x is greater than −1 and at most 3.

Graphing Linear Inequalities in Two Variables

For y ≤ 2x − 1:

1. Graph the boundary line y = 2x − 1 as a solid line (≤ includes the line).
2. Test a point not on the line, often (0, 0): 0 ≤ −1 is false, so shade the side that does not contain (0, 0).
3. Use a dashed line for strict inequalities < or >.

USTET may ask which point satisfies an inequality rather than requiring a full graph: plug the coordinates into the inequality and check truth.

Connecting Systems and Inequalities

A system of inequalities asks for the overlapping shaded region. On a multiple-choice exam, the fastest route is usually testing each answer choice against every inequality rather than sketching carefully. If a choice fails even one inequality, eliminate it.

Example: which point satisfies both y ≥ x − 1 and y < −x + 4?

Test (1, 2): 2 ≥ 1 − 1 → 2 ≥ 0 true; 2 < −1 + 4 → 2 < 3 true. Point (1, 2) works. Test (3, 0): 0 ≥ 3 − 1 → 0 ≥ 2 false. Eliminate.

Common Traps

• Forgetting to reverse the inequality when dividing by a negative.
• Treating a system as solved after finding only x and never recovering y.
• Adding equations that do not cancel anything — then you still have two variables and have made no progress.
• Using a closed circle for < or > (those need open circles).
• In word systems, defining variables unclearly so that a + s = 8 is written but the money equation uses the wrong prices.

Section Takeaway

Choose substitution or elimination to find the unique (x, y) that satisfies both equations. Flip inequality signs only when multiplying or dividing by a negative. For graphs, solid means inclusive and dashed means exclusive. On timed USTET items, plugging answer choices into systems and inequalities often beats lengthy algebraic setup.