Systems of Equations - SAT Math

$(x, y) = (0, 0)$. Add equations: $2x = 0$, so $x = 0$, then $y = 0$.

$(x, y) = (1, 3)$. Set equal: $x + 2 = -2x + 5$, solve to get $x = 1$.

$y = 3$. Substitute $x = 5$ into first equation directly.

Infinitely many solutions. Second equation is twice the first, creating identical lines.

$(x, y) = (2, 1)$. Add equations to eliminate $y$: $3x = 6$, so $x = 2$.

Infinite solutions (dependent system). Second equation is twice the first, so they represent the same line.

Determinant $\neq 0$. Non-zero determinant ensures coefficient matrix is invertible.

Infinitely many solutions. Second equation is twice the first, creating coincident lines.

Use matrices and row operations or inverses. Convert system to matrix form and solve systematically.

Equations represent the same line. One equation is a multiple of the other.

No solution (inconsistent system). Parallel lines with different y-intercepts never intersect.

Lines intersect at one or more points. Consistent means the system has at least one solution.

$x = 4$. Substitute $y = 0$ into equation: $5x = 20$.

$(x, y) = (3, 0)$. Add equations to eliminate $y$: $6x = 18$, so $x = 3$.

Infinitely many solutions. Second equation is $-2$ times the first, so lines coincide.

$(x, y) = (4, 3)$. Add equations to eliminate $y$: $4x = 16$, so $x = 4$.

Non-parallel lines (different slopes). Different slopes ensure the lines intersect at exactly one point.

$(x, y) = (6, 1)$. Add equations to eliminate $y$: $2x = 12$, so $x = 6$.

$(x, y) = (3, 1)$. Add equations to get $2x = 6$, so $x = 3$; substitute to find $y = 1$.

Infinitely many solutions. Dependent equations represent the same geometric line.

$x = 6$. Substitute $y = 0$ into equation: $2x = 12$.

Lines are parallel, same slope, different intercepts. Lines never intersect when they have identical slopes.

Infinitely many solutions. Second equation is twice the first, so lines coincide.

$(x, y) = (-\frac{2}{3}, \frac{5}{3})$. Set equal: $2x + 3 = -x + 1$, solve to get $x = -\frac{2}{3}$.

Add/subtract equations to eliminate a variable. Multiply equations to make coefficients opposites, then combine.

Same slope and same intercept. Lines overlap completely when all coefficients are proportional.

Infinitely many solutions. Second equation is twice the first, creating identical lines.

$x = 4$. Substitute $y = 0$ into first equation: $3x = 12$.

$(x, y) = (6, 4)$. Add equations to get $2x = 12$, so $x = 6$, then $y = 4$.

Infinitely many solutions. Second equation is twice the first, so lines are identical.

Plot equations, solution is intersection point. Draw both lines; they meet at one point.

Solve one equation for a variable, substitute in the other. Express one variable in terms of another from first equation.

$(x, y) = (6, 4)$. Add equations to get $2x = 12$, so $x = 6$, then $y = 4$.

Add/subtract equations to eliminate a variable. Multiply equations to make coefficients opposites, then combine.

$(x, y) = (-\frac{2}{3}, \frac{5}{3})$. Set equal: $2x + 3 = -x + 1$, solve to get $x = -\frac{2}{3}$.

$(x, y) = (4, 3)$. Add equations to eliminate $y$: $4x = 16$, so $x = 4$.

$x = 4$. Substitute $y = 0$ into first equation: $3x = 12$.

$(x, y) = (6, 1)$. Add equations to eliminate $y$: $2x = 12$, so $x = 6$.

Same slope and same intercept. Lines overlap completely when all coefficients are proportional.

Infinitely many solutions. Second equation is twice the first, so lines are identical.

$x = 4$. Substitute $y = 0$ into equation: $5x = 20$.

$(x, y) = (1, 3)$. Set equal: $x + 2 = -2x + 5$, solve to get $x = 1$.

Infinitely many solutions. Dependent equations represent the same geometric line.

Infinitely many solutions. Second equation is twice the first, creating coincident lines.

$x = 6$. Substitute $y = 0$ into equation: $2x = 12$.

$(x, y) = (3, 2)$. Add equations to eliminate $y$: $2x = 6$, so $x = 3$.

Solve one equation for a variable, substitute in the other. Express one variable in terms of another from first equation.

$(x, y) = (3, 0)$. Add equations to eliminate $y$: $6x = 18$, so $x = 3$.

Lines are parallel, same slope, different intercepts. Lines never intersect when they have identical slopes.

Infinitely many solutions. Second equation is twice the first, creating identical lines.

Infinitely many solutions. Second equation is twice the first, so lines coincide.

Infinitely many solutions. Second equation is twice the first, creating identical lines.

Use matrices and row operations or inverses. Convert system to matrix form and solve systematically.

Infinitely many solutions. Second equation is $-2$ times the first, so lines coincide.

Determinant $\neq 0$. Non-zero determinant ensures coefficient matrix is invertible.

Equations represent the same line. One equation is a multiple of the other.

$y = 3$. Substitute $x = 5$ into first equation directly.

$(x, y) = (2, 1)$. Add equations to eliminate $y$: $3x = 6$, so $x = 2$.

Plot equations, solution is intersection point. Draw both lines; they meet at one point.

$(x, y) = (0, 0)$. Add equations: $2x = 0$, so $x = 0$, then $y = 0$.